I wrote this up for a frist-time triathlete. It's enough of a decent introduction, that I thought I would put it on my site for future aspiring triathletes. I realize this isn't that well-organized, so I'll probably update this over time.
This is usually the more hectic of the two transitions. Your brain and lower body will be low on oxygen/blood flow. Your feet are wet, and usually covered in sand/grit/gravel.
Right when you get onto the beach, take off your goggles and swim cap, press the lap button on your watch, and walk or jog to the transition area. Find your bike, and clean/dry your feet enough to get them into your socks. Toss on your shoes next, then your (shirt,) helmet, and sunglasses. Walk/jog your bike to the mounting area (outside of the transition area), press the lap button, hop on the bike, and take off. I usually wait to start taking fluids until I'm clear of T1, and out on a good straight-away. Then, take a few pulls of water to recover fluids lost while swimming.
This is much simpler transition, but your legs will be a little tired, so expect a little warm-up period as your legs adjust to the new stress.
Before you get close the transition area, grab a few swigs of water. While you're slowing/coasting to the dismount area (outside of the transition area), take some time while you're still on the bike to stretch out your lower back and hip flexors. Try to stretch your calves, and relax your legs as much as possible. Carefully approach the dismount area, hop off the bike, press the lap button, then walk/jog your bike to your position on the rack. Hang up the bike. Remove helmet. Switch shoes (if wearing cycling shoes). Grab bib (if wearing number belt). Don running hat (if necessary). Run to the running exit (remember to hit lap button). RUN!
The first triathlon is really about acclimating to the (unusual) race environment. It's great to push yourself, but consider this a learning experience, and something to be proud of when you finish regardless of time.
Your race packet will have a course map. Get roughly familiar with it, but don't worry about memorizing everything. The volunteers will keep you on track. The only exception to that is multi-lap routes. You'll be responsible for counting laps, and you'll get DQ'd if you mess that up (I bring this up since I have been DQ'd from a duathlon for that reason; counting to four is hard).
Some races will give you a chance to hop in the water before the race to "warm up" (or "cool down" depending on the weather/water). Always take them up on the offer. Submerge yourself, get comfortable in the murky lake silt, and swim out to a close buoy and back. It makes a huge difference, and calms your nerves considerably.
For your first triathlon, plan to arrive at the parking lot at least an hour before your start time, or 45 minutes before the transition close time (whichever is earlier). You'll have a few last-minute snags in your plan, and you don't want to rush things. They will shut down access to the transition area at some amount of time before the first swimmers are sent out (the details will be in your race packet).
Take your time setting up your transition "layout." Make things easy to grab during the transition (bike stuff near the close edge of your large towel, running stuff near the far edge). Some people will pre-roll their socks so they can just slip their toes in, and roll the sock up their feet (easier than pulling them over wet skin). Have a look at how other folks are setting up around you.
After getting set up, visualize the steps you'll need to take during each transition. Take the time to walk into the transition area from the swim entrance, and the bike dismount area. Get familiar with your surroundings when there's no stress or lack of oxygen.
Even with staggered starts, you'll still be swimming in close proximity to others. This can be extremely stressful your first couple of races. Keep in mind, the other folks are just as stressed out about it as you are. Be comfortable with quickly sighting landmarks (buoys, hills, peers, boats, lifeguards, the swim exit flags/arch, etc). Until you know how well your swim stroke can keep you going straight, sight about every fourth stroke cycle. Also, there's nothing in the rules saying you can't start at the back and swim wide around the corners. You may end up overall faster that way just from being more relaxed rather than getting tangled in a group of arms and legs.
If you run into problems on the swim: turn over, and float on your back while you catch your breath. If that's not helping, raise your hand, and keep it up until a lifeguard or boat gets to you. Grab on, and take a break (you're still in the race). You can also just change up your stroke. I revert to a lazy backstroke if things just aren't going my way. It's much harder to navigate, but there's no stress. Some folks will revert to a breaststroke which is easier to navigate, but takes a little more energy.
I'm not a fan of eating anything during a sprint-distance triathlon. Your body can't metabolize anything that fast. If you feel the need, make it something simple and sugary that you eat while you're on the bike so you'll have a little sugar high when you're nearing the end of the run. Don't over-eat or "carbo-load" the day before or morning of the race. It's easy to think you'll need a lot of excess calories, but sprints are so short, your body won't need anything out of the ordinary to exert the necessary effort. I have a normal dinner the night before, and a light (no milk) breakfast at least two hours before the race start. Hydration is a lot more important than exotic nutrition plans.
Finally, enjoy the experience. Soak up the supporting cheers from the volunteers and the spectators. Encourage the other racers while you're out on the course (a lot of them will be shouting encouragements to you). This really is supposed to be fun. It's like a silly race a bunch of kids would think up. Have fun!