First song i learned was I'll Follow You Into the Dark by Death Cab For Cutie but i really like playing It Will Rain by Bruno Mars recently
Best songs to play on uke, well for starters just play Lateeya songs, if you are into songs with 5 or more chords I prefer you play Aj Rafael songs, Passion songs or some other artist that uses a lot of chords. Although if you are a picking player then play Eric Clapton songs, and Jack Johnson songs.
I like the sounds I can make, and since I can't sing I have to depend on the Uke. So I am working on pick/strum patterns for different songs that I also enjoy. I have a collection of over 500 songs on cd's, and another 3,000 or so now on the computers. Favorites are songs like, 'Dust in the Wind', 'Hotel California', 'The Unicorn Song', 'City of New Orleans', 'For the Good Times','One too Many Mornings', and most older blues songs and train songs.
I actually really like playing "surprising" covers on the ukulele. I take chords for hip-hop or hard rock songs and make them sound soft and sweet on the ukulele. For example, Hey Ya by Outkast has a chord progression of G C D E. I play the E major chord as an E minor. It gives the song a whole new feeling.
I also like playing sad, slow songs on the uke, especially Bon Iver. Michicant was one of the first songs I learned, and it's also one of my favorites.
Folk songs are great on the ukulele, because they use simple chords, and because the instrument is so simple in itself, it lets you focus on strumming, and adding personal flavor.