With a 1-row diatonic accordion, you have the same notes as a harmonica. With 2 rows (some older Hohner club models, for example), you have enough to play simple songs, with a few chords. If you find a 3-row chromatic (some austrian or slavic button boxes), you'll essentially have all the notes, and it gets easier to play complete songs except you're pretty stuck in a couple of keys. Some Swiss diatonics have 2 or 3 rows, plus an extra row with some sharps and flats that are not in the same key. Still pretty hard to get modern tunes out of it.

To me, a diatonic is best used as accompaniment. If all you need is a couple of chords, it's very simple. If you really want to play accordion, get a chromatic. It's a little harder to get started because it looks like a lot of buttons, but ultimately you'll be able to change keys and make 4 or 5 note chords much easier than on a piano accordion. Also, the buttons are close together and much faster than a piano keyboard.