How to get into Field Test mode on an iPhone:

Enter *3001#12345#* and then hit Call.

(via SoCalWUG)

From Mike Outmesguine's email on the SoCalWUG mailing list:

You will be presented with a menu. Strength is shown next to the RX header under the Cellular menu. And it looks like strength is shown as -dBm. For example, -80 is a good signal, while -110 is poor. The higher the number (negative number, closer to zero) the better the signal. A strength of -1 means you are sitting on the tower and need to move away before you or your phone melts.

On the iPhone, Field Test Mode also shows you something like 100 other parameters (I didn't actually count). And I don't know what a lot of it means... yet. The neat part is how it shows you the cell towers to which the phone is connected... the highest signal strength is listed on top (and hopefully that's the one you are using).

If you don't have an iPhone, here's a list of field test modes for your phone:
https://wilsonelectronics.com/Files/PDF/PhoneTestModes.swf