Does applying vaseline to a cut prevent scarring? If not, how can I prevent cuts from scarring?

No but may lessen it. The degree of scarring depends mostly on the severity of the cut. The more deep and ragged the worse the scar will be.
Medical glue. Scarring has to depth of wound, but medical glue available over the counter can act like stitches to narrow the wound. Also depending on the area decreasing motion to the area will decrease forces which can widen the wound. If raised scar tissue starts to develop after 6 weeks scar creams such as mederma or bio oil can flatten scars.
No smoke or sun. Want to get have that cut heal really well? Avoid smoking (horrible for wound healing) and don't go tanning or have long exposure in the sun. Set your body's self-healing mechanism up for success!