You can buy apps either through iTunes or on your iPhone; to buy on the phone using AT&T's network, the app must be below a certain size threshold (I believe it's 10 MB or so). You can buy apps of any size using your phone if you're connected via WiFi.
The cost of the app is paid through iTunes regardless of the method you use to purchase it (if you have gift card funds available, it will come out of those, and if not, it'll be billed to any credit card you might have associated with your iTunes account.) If you buy it using iTunes, it will be synced to your phone the next time you sync. If you buy it on your phone, it will be synced back to iTunes the next time you sync. You'll end up with the app in both places no matter what.
Have fun with the new gift! I'm a complete and total iPhone nerd so if you ever have questions, fire away. Yes, I'm one of the people who was incredibly excited to follow Engadget's live blog updates today during the media event Apple held to discuss details of iPhone OS 4.0, which will be released this summer (theoretically alongside a new iPhone! I am insanely giddy about that.

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You might also check out some free or "lite" versions of apps and try them out before you buy. There are TONS available on iTunes, and I know somebody here has a thread going (on the Budget Board, maybe?) that has compiled a list of the best free apps out there.