Most all Call of Duty games simply use the same matchmaking process that you're used to on consoles (much to the lament of PC gamers), so you probably wont notice much of a difference - you don't need to use the Server browser at all.
However, the option is there in Modern Warfare 3 (and I assume Black Ops 2), and it's really quite simple to use. First, you just have to enable it:
and then you simply go into Servers
in the main Multiplayer menu.
- Along the top - You've got tabs for Internet, Favourites etc. Most of the time, you'll leave it on Internet.
- In the Middle This is the list of servers. You can sort by the columns (such as Ping or Map) to show different options, and scroll down the list.
- Along the bottom - You can add filters to search for games that are not full/not empty and other options. You can manually Refresh the list, View detailed server information (of the one currently selected), add a server to your Favourites list and Connect to the server.
Take note of the name of the server - a lot of these dedicated servers are modded (and will say something about the server in the name). Some that I see pop up a lot are
- Sniper-only modes (it forces you to use a sniper)
- 10x/20x/50x XP servers - For leveling up quickly.
- Unlimited/Longplay Time - these maps either never end, or have extended time periods (like hour-long matches). They sometimes have voting functionality to choose/change maps
As for free/paid servers - some are free, some not. The one's that aren't are password protected. Most dedicated clans will have their own servers set up, and they either have a membership fee, or are run by advertising (and need an invite to join).
As for finding the 'best' server - it comes down to what you're looking for. From my perspective, I'm usually looking for unmodded or longplay servers with low ping, are free, and are either playing Team Deathmatch or Domination. Your tastes may be different accordingly. It changes all the time - the server that's a lot of fun today might have no-one playing tomorrow.