Meeting/finding friends in Stuttgart...I'm an expat

There are many ways of making friends in Stuttgart.
The easiest and best is joining one of the many clubs or courses that are available for every hobby or activity imagineable. Check Volkshochschule, sports clubs, community centres, etc. You'll mostly meet Germans, not just "other transients" - but why would you want to limit yourself to those?
For an English speaking crowd, try Internations, Meetup, or organise a gathering of Expat.com members (I'd support you, but don't have time to do it alone).

Hi, I'm also an American living a long time in the Stuttgart area. I'm a performer and have most of my gigs on weekends, so not a great fit to do a lot of things with someone who works during the week. But I am flexible with my time and live pretty central (Stuttgart-West) so a quick get together after your work in town could be possible. Anyway, if you need any advice you can contact me. As far as other Americans in Stuttgart - I don't know many. There are people with the local US Army facilities but they seem to keep mostly to base or keep a very low profile. There are always some American students at University but unless one is a fellow student then one is not likely to meet them - and they usually only stay a year anyway. Otherwise, there are a couple of Irish pubs in Stuttgart that tend to be a meeting place for English speaking expats but mostly for the early to late twenties drinking crowd.