I'll agree with Gerri about the many ways to explore the island. While we didn't do Capri as a day trip - we stayed on Capri for a couple of nights - it's not a difficult place to do on your own IF willing to do your own research in advance. She has provided a really good link to info about how to get around. We took the bus up to Anacapri but taxi is preferred by other folks. Everything else - other than the chairlift to the top of Mt. Solaro - we did on foot.
Activities to choose are up to your personal interests, abilities and, possibly, weather. You've been given the excellent https://www.capri.net website for ideas. As previously noted, ignore the many touts for guided tours on that site as, with some reading up, you don't need them.
Ferry schedule from Sorrento:
https://www.capri.net/en/ferry-schedule?from=sorrento&to=capri#
To Sorrento from Capri:
https://www.capri.net/en/ferry-schedule?from=capri&to=sorrento#
During high season I don't advise trying to get on the last ferry to Sorrento late in the game. You'll want to pre-purchase your tickets if wanting to push your time to the last possible hour.