You can only be in one alliance, and you will not be able to see any invitations to other alliances when you are in one.
To join an alliance you will need an invitation. If you are not in an alliance you can see your invitations by clicking "Alliance" on the left of your game screen. Try to choose one which is local to you and has a decent number of active players.
If you don't have any invitations you can look on the map and write to players nearby if you like the look of their alliance, and ask them for an invitation.
If you are just doing it for the tutorial quest it may be quicker to found your own alliance, and then look around for something better once you have the reward.
Once you are in an alliance you can leave it by clicking "Alliance" on the left of your game screena and choosing the "Properties" tab, where you will find that option.