I had to replace some rotted walls in my bathroom and ended up replacing portions of the floor as well. Since we were at this point we decide the old peach colored tub had to go. When I put down the new 1/4" underlayment and vinyl sheet do I run this underneath the tub or just up to the skirt? The new tub is fiberglass, or Vikrell I think is the tradename. Anyway the new tub has numerous "pegs" for support on the bottom were the old tub had just 8 and I didn't know how or if this would affect what the tub sat on.

You can install the 1/4" and vinyl sheet flooring first. I dont see a problem with that. (common in modular/mobile home construction) Your drain plumbing will need to be redone to fit the new tub anyways so the 1/4" height shouldn't matter.

Most damage in a bathroom is to the floor at the tub area. This is from stepping out of the tub wet or water getting outside the tub due to a bad shower curtian/doors or just plain splashing about. The water seeps at the joint where the flooring meets the tub skirt, goes unnoticed for years until you have a problem like you had.

If you install the shhet vinyl under the tub your helping prevent this problem. Just make sure you put down a nice bead of silicone caulking at this joint to prevent any water from getting under the tub unit.