I've been to Capri 4 or 5 times. We always purchase an out and back ticket from Sorrento and land at Marina Grande. From there, you should plan to take the funicular up to the town. You'll see a big tunnel for the funicular in front of the docks, but you purchase tickets from a covered area next to a building off the right.
There are restaurants and shops at the end of the funicular. Some are very, very expensive. But it's nice to look.
There are also the Gardens of Agusto which are very beautiful. On the way there is a granita stand. We just passed by it for years before trying some. I don't know what sort of magic they put in that stuff (or if it's just the effects of the heat and sun) but that is the most refreshing thing I have ever had. Last time I got two - one orange and one lemon.
That is usually enough activity for us. But, there are also ruins of a Roman villa (Villa Jovis) and the Blue Grotto, weather permitting. I know that you can book a trip to the Blue Grotto, but it's not for the faint of heart. We had to step from the boat that took us to the grotto opening to a row boat in the middle of the ocean. You just jump/step and hope for the best. Then you have to lie down completely flat to get under the entrance. It's pretty exciting, then beautiful inside.