Well, first of all, don't go nuts, don't spend tons of money on new things, take it slowly at first, use what you already have, as much as you can.
Buying things does not make you faster, stronger or fitter
Training smart makes you faster, stronger and fitter. Getting off the sofa, getting out to the pool, the track, the park, the road, the trails or the hills is what makes you faster, stronger and fitter.
Triathlon can be an expensive sport, with three different sports that all have their own equipment, plus the growing list of triathlon-specific equipment that is available out there. Triathletes (myself included) can be pretty gadget-obsessed, too, and always seem to be looking for something new to spend their money on. I'm not sure why, maybe it's because we are trying to be good at three different sports, and we often feel a little out of control, or perhaps even inadequate, and so we have to measure everything, quantify everything, in the hope that it will make us feel more in-control. Anyway, I would counsel extreme caution in your spending if you are new to the sport and wanting to try it out, until you are sure that you love the sport. Then, over time, there are plenty of ways you can acquire more and spend more later on.
Think about your equipment well in advance of your race, and give yourself at least several months to get used to your equipment that you plan to use in your race. You definitely do NOT want to go out and buy, borrow or rent new and unfamiliar pieces of equipment a few weeks away from your first race, when you won't have enough time to get used to it. Familiar, comfortable equipment is what you want on race day. Also, think about your nutrition and hydration (more detailed, full-length blog on that subject later, but for now, keep it simple). For example, how will you take water or a drink with you on your longer rides and runs of more than 45 minutes? What will you eat and drink for those longer sessions? There is a myriad of advice for and products related to these issues and you want to give yourself plenty of time to find the methods and items that work best for you. Again, don't leave it until race day (or a week or two before race day) to start thinking about these things.
What equipment do you use when you go swimming now? If you already own a swimsuit, swim cap and goggles, and if your first triathlon starts with a pool-based swim, then you already have everything you need for the swim section of your first triathlon. If you don't normally swim with a swim cap, go get one and start swimming with it now - swim caps are going to be required in almost every triathlon you ever do, so start practicing with it now. If you have a favorite pair of goggles, then go out and buy a second pair of the same exact goggles right now and save them for later, as a back-up. If you don't have a favorite pair, then try a few different ones out until you find ones that are comfortable on your face and do the job you want them to do.
If your first triathlon requires you to wear a wetsuit, then you will need to go and get one. Typically, in open-water swims with water temperatures between 13 degrees C and 23 degrees C, wetsuits will be mandatory; above 23 degrees, wetsuits will not be allowed; below 13 degrees, swims may be canceled or shortened; in swimming pools, generally, wetsuits are not allowed in competition. You might be able to borrow one from a friend, rent one, or buy one second-hand if you want to save some money here. But, bear in mind that probably the single-most important factor in choosing the right wetsuit for you is the fit. It should feel like a second skin, snug and tight-fitting, not letting any water in (which is why surfing wetsuits or other wetsuits not specifically for swimming are probably not going to be good for this purpose,they don't usually fit snugly enough and let in a lot of water) but also not so tight that it restricts the movements of your arms or pulls down uncomfortably on your neck or chest or shoulders. The way you put on your wetsuit and adjust the material so that it is gathered up in all the right places will also affect the fit, comfort and function of the wetsuit. So, however much money you spend on a wetsuit - or if you borrow, rent or buy second-hand - you really need to be able to try it on and get some knowledgeable help adjusting it before you walk away with it and go swimming in it.
Try and get your wetsuit sorted out several months ahead of your first race, and get some practice in it whenever you can (even if you just wear it for 10-15 minutes in a swimming pool, then take it off when you feel uncomfortably warm). You will be surprised at how different you feel when you swim in a wetsuit, so start getting used to that as soon as you can.
This is the area of triathlon where most people find they can spend the most money. I would strongly recommend that you start by using what you already have. Of course, if you only have a single-speed bike or a fixie, and your first race is going to be on a course with some hills, you may want to consider getting a bike with gears (although a friend of mine did his first race on a fixie and was absolutely, blindingly fast out there, but he was a very strong and confident cyclist long before he got into triathlon).
If you already have a bike with gears, like a hybrid bike, a mountain bike or a racing bike, there are some simple things you can do to make your bike faster and more race-ready, without actually going out and buying a whole new bike for your first race. I used a pretty old mountain bike for my first year of doing triathlons and I had loads of fun, no one laughed me off the course, and I was not the slowest person out there on the bike course, either.
You can put some new tires, with thinner, slicker treads on them (instead of thick, knobbly treads) and a mountain bike or commuter bike will instantly be much faster, for perhaps only a £50-60 investment.
Get your bike serviced, chain cleaned, brakes checked, gears adjusted, and your bike will go faster still, for perhaps another small investment.
Get a friend who knows a lot about bikes to have a look at your bike position and help you perhaps to make a few adjustments to your seat and handle bars to make sure you are sitting in a near-optimal position, then you will probably be more comfortable, more efficient, and even faster on your bike. Even if you have no expert bike-fit friends and you pay for a professional bike fit, it is a lot less than buying a new bike.
You can put toe clips on the pedals of your bike and get a lot of the same benefits you see people getting when they have those weird-looking (or cool-looking, depending on your perspective) cycling shoes with the clipless pedals, but without having to get used to those weird (or cool) pedals and walking around like an arthritic duck when you are not on the bike. Toe clips will hold your feet in the right place on the pedals for optimal pedaling power, stop your feet from slipping off the pedals if you are pedaling very fast, and even allow for some amount of pulling up on the pedals, like when climbing uphill. Plus the extra benefit is that you don't have to change your shoes when you get off the bike and go on the run, as you will already be wearing your running shoes on the bike. PLUS, toe clips only cost £10-20. Of course, it takes some time to get used to toe clips (arguably, not as much time as getting used to cycling shoes and clipless pedals), so give yourself plenty of time to get used to them, too.
You will need a helmet, that is an absolute and immutable requirement for every single triathlon you ever do. If you don't have one, or you are not used to riding with one, then go get one and start riding with it as much as you can.
What clothes will you wear on the bike? Remember, you are going to come out of the swim and jump on your bike, and you can't change your clothes completely in transition (like, you can't get naked or anything). So, do you swim in your cycling clothes? (That's actually what I did in my first year of racing...) What will your cycling clothes feel like on the bike when they are soaking wet? Or, do you jump on the bike in your swimsuit? What will the swimsuit feel like on the bike, without any of the padding you might get from a good pair of cycling shorts?
This is one, new equipment purchase I would totally recommend for the first-time triathlete: a trisuit. There are some pretty good ones that start around £50. You can swim in it, cycle in it (it will have a little less padding than normal cycling shorts, so that they dry reasonably fast but still provide your butt with some nice padding on the bike), and then you can run in it, too.
Running arguably requires the least amount of specialist equipment in a triathlon and, statistically, the sport that most triathletes come from. Chances are, you already have a pair of running shoes that you like and train in. Obviously, if you don't, then go out and get a pair of running shoes and get started!
The best recommendations I have seen (and what I practice in my own running shoe selection) is: get a pair of shoes that feel comfortable when you run in them. Video analysis of your feet when you are running, like they offer in some higher-end running stores to determine what kind of shoes you should buy, is pretty much worthless. (Should we do a video analysis of the tires on your bike to determine how to fit your bike? Duuuuuuh! No! We should not! So, how does a video analysis of just your feet tell you anything useful about your whole running style?) Can we categorize everyone into "overpronators", "neutral" or "supinators" and make intelligent running shoe recommendation based on those over-simplified categories? Well...you can try, but don't use that nonsense, pseudo-science on me in an argument for getting me to spend £100 on a pair of shoes.
Another purchase I can recommend you make in this area, as you prepare for you race day is a set of "race laces" or elastic laces for your running shoes that don't require tying up when you are in transition. The usually only cost about £10-20 and can help make your transitions much smoother in your first few races. Of course, get them well in advance of your race and get used to wearing them on your shoes when you go running.
You may also want to try running without socks on. A lot of triathletes find that it saves them a lot of time in transition and, for sprint and Olympic distances, they can get used to running without socks if their shoes fit well and they practice for a few months before race day.
And, another low-cost item you can purchase that will help make things smoother is a race belt. They cost about £10 and allow you to keep your race number in the right place while you are cycling and running. If you don't have a race belt, then most race organizers will provide you with some safety pins, and then you will have to pin your race number to whatever top you are wearing for the cycling and the running. So, it's up to you, but it's a lot easier to deal with your race number when you have a race belt.
Other things that you may want to spend your money on, that I would argue will deliver a lot more benefit than new pieces of equipment, are support services. If you are feeling a little overwhelmed by all the training and the time requirements, the dos and the don'ts, then a qualified, experienced coach can be a good investment. Perhaps they can write training plans for you to help you organize your time better, or perhaps they can do some private sessions with you, answer a lot of your questions, and just give you the right nudge in the right direction.
Swimming lessons (ideally, with a triathlon-specific, experienced coach or company) can be incredibly helpful. And, if they offer video analysis, your learning process will be accelerated exponentially.
Also, you could consider joining a triathlon club. I have worked with most of the tri clubs in London and most of the sessions I have coached or trained in have been populated by friendly, welcoming, helpful people who love nothing more than befriending and supporting newcomers. We have all been where you are now. We still make some of the same newbie mistakes that we used to make, no matter how long we have been doing the sport. And, we can all be pretty light-hearted and self-deprecating (when we are not being gadget-obsessed, competitive jerks). I know a lot of people are intimidated by the idea of joining a club if they haven't yet done a triathlon and don't feel that they are "good enough" to join in, but you would be surprised by just how wrong that thinking is. Clubs can be enormously valuable sources of support and information, and insanely good value-for-money.