How do I put up a travertine tile backsplash?
For a backsplash, it's fine to tile directly to the drywall without anything behind it. For travertine, you probably want to avoid mastic. Mastic should NEVER be used on any natural stone because it is an adhesive and will discolor the stone (the adhesive has chemicals in it that will leach out over time). Use unmodified thinset. You'll also want to use a non-epoxy and unmodified grout (no latex additives) -- sanded or unsanded is fine. When the whole thing is done, you will want to seal it very well.

Travertine is VERY trendy right now, but you should be aware that it's not a renewable resource (it's mined limestone formations that accreted over thousands of years), the quality varies drastically, it is very porous, and it stains very easily. These things all make it close to my last choice for a kitchen backsplash. Even with a good sealer, it will still retain bits of food that hit it and will stain very easily if you splash something like tomato sauce on it. The sealer needs to be reapplied annually at least; preferably bi-annually. There are many porcelain products that look like travertine but are better suited to a kitchen environment. 

As for how you'll mount it -- Decide on what kind of pattern you want. I have a soft spot in my heart for running bond with travertine. Mix a relatively small amount of thinset. Apply the thinset to the wall using a 3/8" notched trowel and rake it out. Stick the tile to the wall. Wait 24 hours for thinset to dry before grouting. 

Pro tips: 


Use chalk lines on the wall to keep things level to your countertop.
Cut all your tiles ahead of time if you can before you work on an area.  
Mix small batches of thinset, and only work a few tiles ahead. It sets up faster than you think. 
Keep a sponge and a bucket of water handy to wipe up any thinset that leaks between the tiles as you set them in. 
I really like Tavy Tile Spacers because they help you keep the tiles level to one another.