Valve's Counter Strike: Global Offensive is one of the most-played PC shooters on the market, which is partly due to the fact that it was released back in 2012 and so has been sitting on Steam for about six years now. However, if you haven't picked it up during a Steam sale, CS:GO is getting a free-to-play version which will allow you to play against AI opponents and view competitive matches.
In order to get the ability to go online, you will need to purchase the full game. That said, Counter Strike: Global Offensive is on sale all the time. So you can check out the FTP version, and if you like it, pick up the full game on sale.
There is a bit of a double-edged sword to CS:GO's popularity however. The game has a very active esport scene and therefore has some highly competitive multiplayer. In order to do well in online matches, you need to know the maps to figure out where your opponents might be hiding.
Its also dedicated keyboard and mouse (at least on PC), so if you prefer a controller, you might want to give the full version of CS:GO a pass.
You can pick up Counter Stike: Global Offensive's free-to-play version here.