Welcome to CC! I'm sure you'll enjoy your stay.
On your question, there are many solutions. Of course you can buy sets, especially look for sales and deals. You can get individual bricks a few ways:
1. Through lego.com - Once you're there, click 'shop', then 'by category', then 'individual bricks & building accessories', then 'pick-a-brick'. You can go through and get exactly what you want (albeit from a limited selection).
2. At an official LEGO store, they have the 'pick-a-brick wall'. They look like this.
The selection varies over time, but you pay for a plastic cup which you can pack with whatever fits.
. This is a person-to-person community sales site. Individual people have virtual stores where they sell parts. You can really find any LEGO element ever produced. I should point out that various members of this forum have stores, and we are sponsored by Troy's Surplus LEGO
4. Be sure to check out our buy/sell/trade forum. You can post wish lists and what you have, and people may set up trades or purchases with you. The forum doesn't endorse or guarantee individual deals, but there is a thread at the top of the forum that gives feedback for everyone who trades via the forum.
-edit- Okay, I'm an idiot. I could have just pointed you to this question in the FAQ
, which says pretty much what I just re-typed.
One other note. Not to beat a dead horse here, but you'll find that both the grays and brown have changed shades since you were a kid, so bricks that you buy new won't match the ones you have from before. You can get both old and new shades via Bricklink. There were huge debates in community forums like this about the merits of this change, which I won't go into here, but I wanted you to be aware of that.