Depends on your price range. Definately no need to go overboard at first. IMO, its best to ride a cheap bike for at least the first few races. Then if you like the sport and know you are going to stick with it, lay down the money for a nice ride.
If you are new and don't feel like investing a ton of money check the classifieds. BT has them, slowtwitch has them, your local triathlon association may have a classified section on their website and then there is always craigslist and ebay. If you are looking to get in at a cheap price point (you didn't mention what price point that was but I'll assume under 1k) then a used bike will be by far the best bang for your buck.
Also buy a book on bike maintenance so that you are able to keep your older bike running well without having to constantly go to the bike shop. Being able to do the work yourself will save you a lot of money in the long run.
Since you're coming from a mountain bike, the first thing you want to decide is if you're going to upgrade to a road bike, or a specific tri-bike.
Personally, I'd recommend getting a road bike. It'll big a huge step up for you, and getting the fit right on a road bike is much easier than on a tri-bike.
And, there are _tons_ of used road bikes for sale.