I am renovating an old mobile home from the late 60's. I want to install a 2-outlet GFI receptacle and a switch next to each other in the bathroom. The catch is that the outlet has to have constant power, while the switch controls a light fixture in the ceiling (not the outlet). I attempted to wire the new stuff the same way the old stuff was wired, but the outlet keeps tripping the GFI. Another issue I had was that when I thought I had it working, the led light would come on only when a cord was plugged into the outlet. HELP!!

What "main wire"?

There should be a feed coming into the box with a hot, neutral, and ground.
The hot goes to both devices.
The output from the switch then goes to the hot of the fixture, and all the neutrals are connected together (one from the fixture, one from the GFCI, one from the feeder).
All the grounds are also connected together, along with any switch ground pigtail, the GFCI ground pigtail, and a box ground pigtail (for a metal box).