How to monitor signal strength?
There are a some apps available which would answer your question. The probably best ones include...

NoSignalAlert, which not only alerts you when you lost the signal, but also provides a log1 of those events, and lets you even browse those "dead zones" on a map and display the collected log information along2

 

OpenSignalMaps, offering similar features, and giving you the ability to show nearby cells3 (so if your signal is low, and you want to have a conversation, you can move in the right direction4 for a stronger signal)

 

If you are primarily interested in mapping signal quality, Antennas might be worth a look as well5. It can export collected data to KML, so you can display it in Google Maps / Google Earth on your computer as well.



Then there also is already mentioned RF Signal Tracker, again rather interesting for mapping6, which additionally tracks Wifi signal

 

Last but not least: G-NetTrack is also good for mapping8 of daily routes and the like.





What might be of even more interest to you is how to automatically react on a weak signal in order to save some juice. If so, I recommend you taking a look at the question What is Cell standby and how can I keep it from eating my battery? (and of course its answer).