what size trowel should I use?
1/4 or 1/8th inch notched trowel.  Really it depends on how thick the tiles are.  If they are thick mosaic tiles I would go with 1/4th inch.  If on the thinner side you want 1/8th.  

You want good coverage but this is just a backsplash.  So looks and getting too much thinset in the cracks trumps the other stuff.  If I had a perfectly flat wall I would almost always use 1/8th.  You can give yourself more wiggle room with 1/4th but this also may make your install take longer.