Question about tub/shower sealing

I'm one of those paranoid owners that will frequently look under the cabinets and bathtub where water connections and drains are located just to "have a look" to make sure all is well. Yesterday I discovered a slightly moist area on the floor under the tub. The area was where the back wall meets the front wall at the front of the tub. It is very slight and appeared after I took a shower. My tub and shower are not one piece, and there is a seal where the tub meets the walls (covered in linoleum like the floors) that is attached with screws. The inside corners of the walls have a piece of trim.

I am going to reseal the tub where it meets the wall, and where the walls meet. Before I get started on this, I would appreciate any advice on how to remove the trim pieces, and what may be involved in resealing. It appears pretty straight forward, but I've never done it before. Thanks for any advice.