From what I recall, Uncle Rod's Boot Camp is based around the most common chord progressions; millions of songs use only that, or can be played with just that (though eventually you'll want to add more spice to them). The boot camp takes you through the main chords in several keys—these chord sets have the same relations between them, so the same songs work with most of those keys, you just have to switch the chord set according to the key. If course, when you change key, the song may not fit as comfortably with your voice (or, conversely, it may fit better).
There are lots of lists floating around of 3- and 4-chord songs. I ran across some at Mudcat Café:
http://mudcat.org/
Easy songs for uke and guitar:
http://www.mudcat.org/thread.cfm?threadid=147110#3408561
Three-chord songs:
http://www.mudcat.org/thread.cfm?threadid=14219#121112 (You'll have to sift through a lot of blather.)
There are some massive collections of songs which you might get from the library and leaf through. If you learn how to transpose (using, say, a double wheel of fifths or a transposition chart), you'll find a slew of songs that only use a limited number of chords, and that they tend to have the same relationships as the set of chords you're learning, just maybe in a different key.
Another source is YouTube tutorials (and not necessarily for uke); this gives you access to a wider set of styles, to songs you're more interested in. It kind of flips the approach: you think of something you want to play that doesn't sound too complex harmonically, you search YouTube to see if there's a tutorial for it, you jot down the chords the teacher is using, which will probably match one of the sets of chords you know, with maybe one or two other chords you need to look up. You may have to transpose the chords—or just move on to another song for the time being. YouTube teachers often take a graduated approach (though the videos in their channel may be presented in a somewhat random order), so if you find a teacher you like who uses songs you like, try to figure out which videos start with the basics and which ones carry on from what went before.