Andrea, those weren't my first attempts at diaper bags. The first one got thrown away b/c I was going w/o a pattern!! I went and bought a Simplicity pattern specifically for diaper bags and lemmie tell ya...having instructions REALLY helped b/c I'm not too good at figuring out things on my own w/o a pattern or at least a starting point!
For the lining, I'm just using cotton quilting fabric from Wal-Mart. I'd love to use something a little more sturdy, like oil cloth, but since I spent so much $$ on the upholstry fabric for the outside, I needed to be reasonable for the lining. Everything is machine washable, though, so the quilting fabric shouldn't be an issue.
And yes, your idea of making two of the bag and then topstitching them together is exactly how I do lining. Same concept when you're lining a dress: make two and sew them together wrong sides together