To buy Forge Points, you just click/tap on the plus sign next to the FP bar. A window will show giving you the option to buy FP with coins or Diamonds.
To invite players that do not currently play, I believe you have to provide their email for Inno to send them an invitation.
To invite current players to be your "friend", you click on their name on the social bar or on the global ranking list and click on "Add as friend". You can also do this when actually visiting a current friend's Tavern by clicking on the avatar of anyone seated there.