Check flashing around the base of the chimney, along the walls and around the plumbing vents. Look for areas where water might be getting in -- gaps, bent flashing, holes made by roofing nails, etc. Flashing is your main defense against leakage, so you want to make sure it is securely fastened and that there are no holes or gaps.
If any holes are found, remove old adhesive or caulk with a putty knife or 5-in-1 tool and wipe down the area. You want to make sure all of the old adhesive or caulk is removed so you can get a good adhesion with the new caulk.
Seal the problem seams with roofing caulk using a standard caulk application technique. Like any other caulking job you want to push the bead, not pull it. You don't have to make it look nice -- you just need to fully fill the gap. Also, there's no need to worry about the roofing cement getting wet before it dries. It can be applied to a wet surface and it will cure, even in the rain. Fill any nail holes with the caulk and smooth with a putty knife.
Check rubber collar around the plumbing vents. If they are worn or brittle, replace them with new collars. You also want to check the flashing around plumbing vent pipes. Some vent collars are equipped with a rubber sleeve that fits tightly around the vent pipe. These can become brittle from exposure to the elements. Feel the rubber to check if it is hardened or brittle -- if it's deteriorating, replace it.