I exercise at home or a gym. I don't exercise the day after a tattoo, but start in the 2nd day after a tattoo. As for the sun, I would just keep it out of the sun. It won't kill you to stay out of the sun for a week. The sun isn't good for your skin under any circumstance, but I wouldn't subject freshly tattooed skin to it. My dogs are pretty mellow, but I keep them away from the tattoo for a week. I wouldn't want them rubbing against it or god forbid, licking it this early.
I'm probably overly cautious about my tattoos, but it's one of the reasons they all look new after 13 years.