1/16, and thinset would be the best route.
www.johnbridge.com is a great source of info on tile work.
Doing a tile surround with 4X4 tiles. Do I use 1/16 spacers or 1/8 spacers ?
Thinset or tile mastic ? Over 1/2 inch hardibacker
Not all tile are created equal...if your tile are not very consistant (gauged), they can vary in size from one to another. A 1/16" spacing is tough to keep the grout lines straight - a larger one may be needed. 1/8" gives a little more leeway. Some wall tile have a built-in nub to set the gap, and no spacers are needed.
If the tub is also used as a shower, use real thinset - the stuff you mix with water; never use mastic (even if it is labeled thinset!). Mastic, if it stays damp, will turn mushy. Thinset, being cement, gets hard and stays that way. Real thinset is usually cheaper, too, sometimes by a bunch. Caulk wherever there is a change in materials or a plane change (i.e., corners, and at the tub/tile junction), do not grout those junctions. www.johnbridge.com is a great place to get tiling help.
Important note - I'm not a pro
Retired Defense Industry Engineer; Schluter 2.5-day Workshop Completed 2013, 2014