Remove handles/pulls/knobs from your vanity.
Measure your trim boards. The easiest way is to line up each piece where it will go on the cabinet, starting in one corner, then make a pencil mark on the back where it overhangs so you know where to cut it off. Do that for each piece now.
Using the miter box (or a miter saw if you have one), cut the corners of each piece at a 45 degree angle.
Use construction glue to glue each piece of trim to the face of the drawers or doors, following the instructions for your glue.
Use wood filler to fill in any holes from the previous hardware (if applicablle).
Using paintable caulk, put a thin bead of caulk everywhere the new trim pieces meet the door/drawer. (So, along each edge.) Dip your finger in water and smooth out the bead of caulk. Let it dry according to the package instructions.
Prime and paint your cabinet as needed. If it's laminate, I recommend 2 coats of primer.
Add new pulls/knobs/handles.