Lily, I'll be perfectly honest and advise against trying to do this.
First off, there is no "fast" way to Capri from Rome.
Secondly, even if you put in the time and effort it's going to take to get there, there's no guarantee that you'd even see the grotto: it can't be accessed when the seas are too rough, and that can occur at any point during the day. Chances are fewer than more, for sure, but it happens enough to be a concern.
Thirdly, it would involve a LOT of time and money for the very short amount of time you'd actually be in the thing even if the seas DO cooperate. It would be a lot of time that, IMHO, could be much better spent actually sightseeing elsewhere and which would involve little-to-no risk.
Capri really only makes sense as a day trip from Naples or another closer location, and I'd never do it if the grotto was the only reason for going.
That said, probably the quickest way would be fast train to Napoli Centrale, taxi or bus transfer to the port, jetboat to Capri, and then bus or transfer to the grotto. You could also take an urban train from Napoli Centrale to Molo Beverello port - if you choose a boat which departs from there - in Naples. Check train schedules/prices on Trenitalia or Italo:
http://www.trenitalia.com/tcom-en
http://www.italotreno.it/en
There are no direct or high-speed trains to Sorrento from Rome; you'd have to change to a local Circumvesuviana train in Naples if wanting to get there via rail.
Check boat schedules from Naples to Capri (or Sorrento or Capri) here:
https://www.capribooking.com/en/ferry-schedule?path_id=11
Buses run to the Grotto from Anacapri on a regular basis or book a direct transfer from the Capri marina.
http://www.capri.com/en/s/la-grotta-azzurra