Common songs such as heard in many places are: (you can search in the DT for the song lyrics)

Black Velvet Band
Fields of Athenry
Molly Malone
Leaving of Liverpool
Shenandoah
Amazing Grace
Will Ye Go Lassie
Loch Lomand
Water is Wide
What do you do with a Drunken Sailor

That's a quick start of some popular songs heard around the English Speaking world. Lots of history you can teach them about in the song or story of the song. Much of it is found in the Forum.