I just got a Dell Inspiron 5160 with 512 MB RAM and 2.80 GHz
processing speed.  It has a 64MB GeForce FX Go5200 video card.  My
question is this: will I be able to play any of the '3D' PC games?  If
so, what's the most recent generation of games that will play on my
computer?  I assume that games coming out this year won't be do-able,
but how about games from '02-'03?  Thanks.

I think you will have no problem, even with the newest 3D games.  Take
a look at the system requirements for Doom 3 here:

http://www.doom3.com/sysreq.asp

It looks to me like you meet or exceed all of them.  Of course, you
can always download a demo and see how it runs.