There were 3 campaigns released with the game, Savanah, Android Rebellion and The Metaborator, each with 7 maps. A small campaign for newbies was released with the 1st patch, Deep Core - if you're confortable with the game, you should probably skip this one. The latest patch, 1.2.1, introduced 2 new player-made campaigns.
In order to play a campaign, you have to first select one of your custom squads, and then in map selection you get the 1st map of each campaign to choose from. The recommended skill level is displayed right next to them.