I feel your pain, believe me! I bet your utility sink looks like this beauty:
This is the only picture I have that shows any of the plumbing under the tub:
Now, almost all of the PVC plumbing in this photo is wrong, and has been replaced, so try to ignore all of the white pipes. My old tub had signs of leaking where the tail pipe connected to the bottom of the wash tub. that piece was brass (or copper) and had 2 bolts that connected it to the bottom, and I believe a gasket. I would check that spot, and the trap connections. I ended up replacing the entire thing, because one of the tubs was cracked, and I had to fix all of that plumbing anyway. This is what it looks like now:
Post some pictures as soon as you can - the members of this forum are incredible, and will get you fixed up in no time!