I actually have an older Galaxy S and I installed CyanogenMod on it without Google's services, so no Google Play, no GMail, etc... The phone is actually usable. I have more apps on it than I have on another Win Phone 8, a Lumia that I received as a gift.
As a browser, I'm using Firefox, which I happen to believe it's the best mobile browser right now. For email I'm using K-9 Mail, a pretty good open-source email client that can also do PGP encryption btw. My contacts and my calendar are synced fairly OK, since I'm connected to my Google account through the Exchange protocol.
For maps I'm using OsmAnd - extremely awful interface, but the maps are from OSM and are awesome and because it helped me get to where I wanted, I have tolerated the UI. For music and movies I'm using Apollo and VLC.
For Facebook and Twitter and other high-profile apps, I could have gone with Amazon's Appstore, however I preferred to use the web interfaces, packaged as apps in Firefox's marketplace. Not as snappy as the native interfaces, but they work and you also get better sandboxing.
The amount of work to go from zero to open-source Android is huge. You could fork Android and release something usable today. Amazon did it ;-)