Erm... buy game, find friends.

It's easier to find friends in games like Left 4 Dead 2 and much harder in common FPS (Counter Strike, Battlefield, Call of Duty).
Some games less suitable for playing with unknown people (cooperative puzzles like Portal 2 or Trine 2).

You may try to find threads where people asking which games are suitable for finding friends. I've seen several here.

There are lots of decent free-to-play games, but that won't allow you to add friends. Spend some of that money perhaps buying the cheapest TF2 item and you'll also become a Premium TF2 account in one step. Of course any actual purchase will then allow you to do all those fancy things like adding friends.