Here are a few tools that I use. If you have one that I havent listed below, please add it in comments.
The screen shot shows the main tools along with my favourite, SpeedTest. While I agree that Speedtest is not the most reliable approach to speed of your connection, I think its the most standard one being used.
WiFi Analyzer is another great app that can be used at home and other locations where people complain about not getting good WiFi speeds. I have been at locations where the 2.4GHz is absolutely packed with APs. 5GHz is also getting busier, though there are still a lot of free channels.
G-NetTrack Lite is a great tool to keep track of the cells you have been visiting. In case you are driving this can collect a lot of valuable info. The paid version, G-NetTrack Pro can collect the info in form of a map that can be used for offline viewing with the help of Google Earth.
I use LTE Discovery mainly for finding the band I am currently camped on. It would be great if a tool can give the exact frequency and earfcn but the band is good enough too. I was once in a situation where I could see two different cells but they had the same PCI. Only after using this, I figured out they were on different bands.
Finally, Network Cell Info Lite gives neighbour cells which can often be useful. I am not sure of these are the neighbours from System Info or from Measurement Control messages sent by network or just something like Detected cells that the phone sees around.
Pind and IPConfig are other tools that can come handy sometimes.
Are there any other tools that you like? Please share using comments.