When I tell people I play the ukulele, I often get the same reaction--a sort of involuntary smirk / laugh. This little instrument doesn't get much respect--at least among people my age (younger folks are more open to it). You can add gravitas to your argument that the uke is a real instrument by playing this piece for your skeptics-- Bach's Sleepers Awake.
Like a lot of Baroque music, it works well on the uke. I play the arrangement by Roger Ruthen, available on his PDF minstrel site (free).This arrangement is easy. You're playing the melody only--nothing fancy. And yet it sounds pretty and natural on the uke.
And after you master this piece, you may want to try some of the pieces in The Bach Uke Book by Rob MacKillop.
Playing fingerstyle ukulele, with information about the songs and where to find the ukulele tablature so that you can play these songs yourself.