Just bought a home, downstairs has a lot of travertine which is in great shape, I just do not like it.
I do not want to remove it, so am thinking of floating over it. Any hints or guidance you could share would be greatly appreciated.

A floating floor over travertine would be the way to go. Use a foam underlayment that has a moisture barrier built in. You will need to undercut the door casing and jambs and install base shoe that covers the expansion gap at the edge of the room.
I use a piece of quarter inch plywood, ripped to a width of about 3/4 - one inch width, depending on the thickness of the flooring, screwed to the wall to act as the expansion gap spacer so the flooring can be scribed to fit and just unscrew it after the flooring has been laid.
Be sure to read through all of the directions on the box.

What kind of underlayment would you suggest?

A closed cell foam, styrofoam, or rag type underlayment will work. You may need to have one with special qualities, like sound deadening, if you are not on the ground floor.