I have had an Android device of some sort or another for over a year now and I love it. Just like most geeks, I use the hell out of it and see what works to the point where it breaks. The good news is it hasn't really broken yet. The bad news is that simply means that I probably haven't tried hard enough.
Here is a list of software I currently use on my Android devices that are more technical in origin.
Ookla Speed test - Great utility if you ever question your data connection's speed. The only thing is that a weak WiFi signal will skew the results.
Net Scan - Need to find a device on a network? Need to scan to see if a port is open? Do you have any idea what I am talking about? Then get this app.
Wifi Analyzer -I really like this one when I travel or troubleshoot WiFi networks. It gives you a visual of the various wireless networks to be found.
Fing - Another tool like Net Scan but it some things better than Net Scan - and some things not at all. Still a good tool.
Teamviewer - Remote Desktop type of app. Love Teamviewer's cross-platform capabilities. Installed on everything I can.
Pocket Cloud - This app does VNC and RDP protocols. Installed for RDP support.
Android VNC - Really good VNC tool.
File Manager - My phone came with a file manager but it pales in comparison to this app.