I started the same way last year. It's pretty normal if you haven't been working out.
I had to start with wall pushes and work from there. Even after a year, I'm still working my way into good shape. Granted, I'm a lot older than you, but it does take time.
Here's a page with a video where Mark demonstrates pushup progressions from wall pushes to more advanced pushups.
How-To: Proper Pushup Technique | Mark's Daily Apple
If you haven't gotten it yet, get the Primal Blueprint Fitness e-book which will walk you through things.
Primal Blueprint Fitness | Mark's Daily Apple