There is a wide range of boat tour prices depending upon the specific type of boat, tour, and departure port. A shared 2-hour boat tour around Capri that departs from the island costs EUR 19/person. This type of tour doesn't include a stop to swim.

You can also opt for a package that includes the boat tour plus funicular tickets, lunch at a local restaurant, bus tickets to Anacapri, and chairlift tickets to Mount Solaro.
Capri Day Tour Package
If you prefer a private boat tour, consider that prices begin at around EUR 160 for a two-hour tour for up 6 passengers.
Take a look at shared boat tour options on this page, or follow these links to view private boat tour options:
Private Capri Boat Tours

Tickets to the Blue Grotto are almost never included in the cost of tours, since the cave can be closed with very little advance warning due to sea and wind conditions. Instead, you'll need to purchase tickets upon arrival at the cave entrance from the floating box office. Only cash is accepted.

Tickets cost EUR 18, including the entrance fee and the cost for the boat service (EUR 12). Children under 6 enter for free.
More information about the Blue Grotto

Tickets for shared tours can be purchased directly at the port, though it's a good idea to purchase them online to save a few euros and avoid long waits during the busiest months of the year like July and August.

Private boat tours must be always booked in advance.

Tours don't only depart from Capri...there are several boats with departures from Naples, Sorrento, or Positano that circle Capri every day, with spare time to visit the island. Both private tours and small group tours are available.

Here are links with options for both private and group tours:

Group tours from Sorrento
Private tours from Sorrento
Group tours from Positano
Private tours from Positano