Play cover songs during your live sets. You can play and cover any song that you want without paying royalties or owning the rights since the performance is covered under fair use provisions of United States law.
Tell the audience what the song title was and who the original performer was. While not legally required, it is a respectful thing to do and it prevents the audience from getting confused over who originally wrote, performed and owned the song.
Take down any online videos of cover performances immediately upon request. It is a violation of copyright law to record and distribute a cover song you don't own the rights to. Live recordings from cell phones and camcorders are common and likely won't spawn lawsuits due to the cost of lawsuits; however, if a takedown request is issued from the copyright holder, take the video down upon receipt of the request and do not repost it.