I'm not an open source zealot, but I generally prefer to use open source applications over proprietary ones if they stack up. Are there any directory sites with reviews, ect that specialize on open source Android apps? The closest thing I've been able to find is Wikipedia's list.
You can install F-droid to get a market-style repository of Free and Open Source android apps:
Google Code seems to be the predominant place for open source Android apps:
This is another source:
As far as my own experience goes, Aptoide is really great. I discovered it when I was trying to find something similar to vshare for the iphone on android**
*.I downloaded the APK Installer for the app directly from their website - aptoide.com.*** I don't really know anything about f-droid, I might try i though. I did try blackmart before and didn't go very far with it, it didn't feel trustworthy. That was awhile ago though.Anyway, I have issues with google play and try to avoid it as much as possible, so I am solely dedicated to aptoide and it hasn't failed me yet. As far as safety goes, I occasionally install software such as AVG and Advanced Mobile Care if I feel I might be taking a risk. I haven't been afflicted with any malware from aptoide's markets yet! On the other hand, my 10 yr. old son is not as critical as I am when he chooses to download something from them on his own tablet, he also refuses to run any software like AMC & AVG and his tablet does have adware.But for as far as it goes , it doesn't seem to be doing any real damage to his device.
You can install F-droid or you can go to fossdroid.com (frontend web of F-Droid) to browser open source apps by trending, popularity, ect... (with screenshots).