In order to find Facebook and Game Center friends, you will first need to be linked with Facebook and/or Game Center in Tunnel Town. To learn how to link your burrow to Facebook and/or Game Center, read this article.
For Game Center:
Once you have linked your burrow with Game Center, you can find your friends on Tunnel Town by launching the Game Center app. With Game Center open, select ‘friends’, then tap on ‘+’ to create a friend request, and enter your friend’s Game Center ID or email, then tap "send". Once they accept your request and install Tunnel Town (if they don't already have it), they will show up on your buddy list in Tunnel Town. They will also appear in the Game Center app itself, under Games-->Tunnel Town.
To visit your friend's burrows, simply click on the "visit" button next to their name and you will be taken to your friend's burrow!
Once your Burrow is linked with Facebook, tap the friend icon on the bottom left side of the screen while in Tunnel Town. A box will appear with "Social" at the top. Any current Facebook friends who are already playing Tunnel Town and also linked with Facebook should automatically show up on this friends list. It will tell you if they are a Facebook friend, or a Game Center friend (If your burrow is also linked with Game Center) by a little icon in their profile icon or picture. It will be either the Facebook icon or Game Center icon.
To invite Facebook friends to play Tunnel Town, tap the "invite friends" icon at the bottom of this page. Another page will appear, showing all of your Facebook friends. Scroll through the list and tap the little envelope icon next to your Facebook friend's names and profile pictures. A little check mark will appear where the envelope icon was. You can invite multiple Facebook friends at once, simply find the ones you want to invite and repeat the process of tapping the envelope icon next to their names. Once you have found all the friends you'd like to invite to play Tunnel Town, scroll back up to the top and tap the "send" button on the top right side. You will then get a pop-up that says "friends invited", just tap "OK" to close this pop-up.
Once your Facebook friends get your request to join Tunnel Town, if they choose to download the app and link it with Facebook, they will appear on your friend list!
Please note that Game Center is for Apple device users only.