Here's another extremely well known song to get you playing your 3 chords. We're not going to try to execute any fancy rhythmic stuff just yet, just work on getting through all the changes, and hum or sing along to whatever strumming you feel like.
As you work on this song, try to notice what you are thinking about while you are playing. Are you concentrating on finding the right chords? On strumming? Trying to remember the words? One thing that you will be striving for as you study the ukulele is to be able to stop thinking about all those things.
Through repetition, all the individual actions will become automatic, and eventually you will think of nothing, only be immersed in the song you are playing. It's like learning to speak - eventually, you aren't thinking about which words to pick, or how to say them, you're only thinking about what you're talking about.