The bikeless triathlete quip notwithstanding, I think you're doing a fine thing. That's an interesting triathlon if the bike portion doesn't go over roads. Spells trouble for a lot of people with their fancy-schmancy tri-bikes.
Anyway, what you'll need is a decent, run-of-the-mill hybrid bike. That's really just a de-tuned mountain bike, probably with no front shock and tires that are fat but not knobby. You won't go terrifically fast, or look like you're on the local pro team, but you'll be comfortable and it'll get you there.
First thing is to figure out what size you are. For a cross bike, find a piece of wood or wooden dowel about a foot long. Stand in your stocking feet with your back to a wall. Shove the stickor dowelbetween your legs and bump it up against your crotch. Now, the trick is to mark the height of that position. Maybe a (close) friend can help you. That's your inseam height. Subtract one or two inches from that, then another ten (the average distance from the bottom bracket to the ground) to give you the maximum frame size you'll need. Obviously, though, the best test here is to go to your local bike shop and road-test a few different-sized models to ascertain what ride suits you best.
So what bike? Lots of choices, certainly. Looking on eBay, for instance, I found a Gary Fisher Nirvana for $300a perfectly good all-purpose bike. Use the keywords "hybrid bike" or "hybrid bicycle." Any bike from Fisher, Cannondale, Bianchi, Trek, or other brand-name maker would be fine. Look for rigs that have Shimano drivetrain components and 26-by-2-inch tires. That's all you need.
In a bike shop, expect to pay around $500 and up for a new hybrid bike. One very good buy is REI's Novara Buzz, running at just under $600 (www.rei.com). Trek's 7500 FX is another good one, in a similar price range ($660; www.trekbikes.com).