you still want a multimeter, however the key to sniffing circuits is having a known reference point. The best place to start is finding ground. Once you have ground (0V) you will have a reference to which you can measure polarity and potential(voltage) of the point your testing. (your negative lead or black lead will connect to ground and red lead will connect to point in question.)
There isn't really a tool you can use to just touch one point and find out what it is, you need a reference to go by......kinda. You can use a current meter setting on your DMM and break the line you are testing in order to see which way the current flows however you still need to know where your reference is. (if that confused you ignore)
finding ground can be a trick, best way is to trace it from your powersource, batteries are easy since negative is usually your ground and it is (usually) a single sided meaning that there is only positive voltage in the system. When you get into AC often times all bets are off espeacially on a two prong device.
Oh yeah, generally you tie your case to ground to help with EMI, so if you see something tying to a metal case from you circuit board, that will be ground. (generally)
Research it online before going through the trouble, a lot of times someone has already done the work for you. Be careful electricity is dangerous.