What is a good, entry-level mountain bike for doing cross-country races/triathlon/cyclocross?

Hard tails are best, unless you are riding on really rough terrain. It's difficult to suggest a bike if you don't give a price. But Trek makes a very nice entry level bike for about $400 in its 4300. Gary Fisher makes some very nice entry level bikes in the $250-$400 range. And the Specialized Stumpjumper can be had for less than $500. I'd stay away from anything you found in a deptament store, and instead head right to one or more of your local bike shops. They can be very helpful in choosing the right bike for your budget. And they don't sell crap that will fall apart in a year. :)

You could get a good entry bike for about 500USD, and a good full suspention for about 800USD You could benefit from a full suspension bike since its gentler to your back as it absorbs any impact. If you plan to do any outdoor riding a full suspension is the way to go, regardless of your riding style. Nowadays Hardtails are mostly used on the competitive level because they are lighter, but the average rider will enjoy the confort of a full suspension.

You could get a good entry bike for about 500USD, and a good full suspention for about 800USD You could benefit from a full suspension bike since its gentler to your back as it absorbs any impact. If you plan to do any outdoor riding a full suspension is the way to go, regardless of your riding style. Nowadays Hardtails are mostly used on the competitive level because they are lighter, but the average rider will enjoy the confort of a full suspension.