I whole-heartedly agree with Christina’s Getting Started tips [see the first entry], especially about gear. Here are a few thoughts to expand on hers. As you remember, she said the most important things are to get a bathing suit, a bike, good running shoes and socks.
The bathing suit part is pretty easy. You’ll want something you can move around in, meaning a “sport” bathing suit that has adequate support. The most obvious choice is a one-piece speedo-type suit or a two-piece sport suit. For later tris, you may consider looking into tri gear like a tri bra (a quick-dry sports bra) or tri tank, and tri shorts (spandex shorts with less padding than bike shorts with a quick-dry feature and side pockets).
For your bike, there are a few things to consider. Since most triathlons will be held on the road, it makes sense to get a road bike. Unless you’re an expert in bike adjustment, you’ll want to go to a bike shop and ask them what size frame you’ll need. (I’m just under 6 feet tall and I ride a 58 cm frame. The seat should be high enough that your leg is nearly fully extended at the base of your pedal stroke.) You don’t need to buy the top-of-the-line bike for a sprint triathlon, but expect to pay between $800 and $1,000 for a new bike. If you’re buying a used bike, make sure to ask the previous owner if it’s ever been in a significant crash. Look all over the frame for signs of damage that could hurt the structural integrity of the frame (these are more than chips in the paint, you’d look for dents in the metal). You'll want to take the used bike to a shop to get adjusted, cleaned and ready to ride. If you're buying new, the bike shop will do all this for you.
Take Christina’s advice on the shoes—go to a shop like RunTex or Rogue Running and let them analyze your stride, how your foot hits the ground, etc., so you know what type of shoe will be best for you.
Socks are easy, too. Get a sport sock that will dry quickly. As you start doing longer distances, you may consider packing two pairs of socks, one for the bike and one for the run, so you have a dry, clean un-blistered foot throughout the whole race.