You won't be able to start GTA Online right from the get-go. That's because you'll need to complete the GTA single-player prologue first. Once you gain control of Franklin, you should be able to switch to GTA Online via the menu or by selecting it when launching the game.

You can launch an Invite Only Session from the single player mode: select Options > Online > Invite Only Session. You can also choose to launch a Solo Session if you don't even plan to invite any friends to your game.