I would use caulk that matches whatever grout color you choose. It's sold in the grout section of any big box store.

Ditto, definitely color-matched caulk. Something like these:

https://www.menards.com/main/flooring-rugs/tile-stone/tile-installation-maintenance/tile-grout-caulk/tec-reg-accucolor-reg-unsanded-caulk-820-10-5-oz/3304786711/p-1444432108046-c-6582.htm?tid=-5222317843907918053&ipos=6

https://www.menards.com/main/flooring-rugs/tile-stone/tile-installation-maintenance/tile-grout-caulk/tec-reg-accucolor-reg-sanded-caulk-850-10-5-oz/3304966711/p-1444432100720-c-6582.htm?tid=-5222317843907918053&ipos=7

Do you also recommend the same color along the base? Or going with a clear caulk on the countertop?

I plan to use the same color along the base. I just installed our backsplash and kept the tile 1/8" off the counter to match my tile spacing.

I used the matching caulk around the entire perimeter of the tile. It was a little hard to apply and tool the joint , but it matches the grout perfectly, including the sheen.

Caulk that matches the grout used. The grout manufacturer will have one (but necessarily silicone, which is more resistant to mold). Any time you change planes you use caulking.