Yes, you are safe using hand sanitizer and some surface sanitizers on good quality medical gloves. I would be wary with the cheaper, food service style gloves.

I would save gloves for use for cleaning and specific handling of unclean or potentially infected surfaces/items. Cleaning your hands well, not touching your face etc will provide the needed protection. Every person I have seen with gloves on in public were getting NO protection from the gloves... touching the shopping carts, things on the shelves in the grocery store, their phone, wallet, car door, steering wheel etc all with the same gloves on! Washing hands, disinfecting surfaces, minimizing touching things in public in general would have provided better protection.