The quick and dirty way to break a signal path on a circuit board is to scatch at the trace with a screwdriver or any other sharp metal object that happens to be handy.
If your new to circuit bending the first thing that I would recommend you do is buy some of these -
Learn a little about soldering/desoldering, potentiometers, switches, and of course safety.
Then take a few of your alligator clips and cut them up then re-solder them back together with a potentiometer and/or a switch on there. Or use a breadboard + potentiometers/switches + 2 wires with alligator clips on one end of them to do something similar without all the soldering. Once you do this you can really dive into circuit bending pretty easily.
The thing I probably use most when circuit bending is a breadboard with an wire+clip as an input + a momentary switch + a potentiometer(or a few different potentiometers of different strength) then another wire+clip as an output just to test things out. Another thing I use alot is an alligator clip attached to a bare wire that I wrap around my finger or something else that is grounded to test out what happens when I ground specific places on a circuit board out.
I hope you understand what I'm trying to say.
And wear eye protection even if you think that you don't need it.
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