I can use
lm_sensors to tell me the tempature of various things on my laptop. And sometimes I can hear the fans starting up. Is there anything that will show me how fast various fans on my system are spinning?
This is working under the assumption that lm_sensors can detect all of your fans. There are a few cases I've seen where the fans are mis-detected, underreported, or not detected at all.
There are several programs to turn lm_sensor fan data into something "displayable". Here's an old write-up from 2003 that describes different programs. Many are still in use today.
KDE appears to have support for this in KSysGuard, although I don't run KDE much so I can't verify if this is in working condition as of KDE 4.0. It appears that this feature is broken for some older releases so I can't speak to its effectiveness.
If you're using a minimalist approach to your desktop, then wmsensors might be the ticket for you. Originally for WindowMaker, it should also work with fluxbox and blackbox.
If you already have lm_sensors installed, there's usually a script you can run as called sensors-detect that will scan your system for devices that report information like that.
Once you run that script (and repeatedly say "yes please scan") you should get a listing of kernel modules that you can load to get more information about your system.
Once you have those kernel modules loaded, you can run the "sensors" command to see the information, and add a desktop widget to show some of that information directly in your toolbar.
If you could tell me what distro, I may be able to give you more specific advice.