You can really see most of Rome on your own without a tour. A good guide is useful in the Roman Forum and on the Palatine Hill, which don't have many signs. The Vatican Museums offer their own tours, or you can rent an audio guide. I have always done the Museums on my own, with the aid of a good guide book, such as the Michelin Green Guide. St. Peter's Basilica also has audio guides.

I don't think it's possible to do a day trip that will include both Pompeii and the Amalfi coast. Some tour companies may offer it, but you will be shortchanging both places. Also, many people have complained that the tours waste time stopping at coral factories or other places where you are pressured to buy stuff, and where they get a kickback. Make sure you know exactly how much time you will have to spend at each place if you take a tour.

It's fairly easy to reach Pompeii on your own by train. You first take a fast train to Naples, then you change to a local commuter train that leaves you very close to the gates of the site.

Smaller shops and restaurants may not accept credit cards, and the use of credit cards for small purchases is not very well accepted. You should keep just enough cash on you for incidentals and a meal or two for a few days, and use a debit card to withdraw cash from ATMs as needed.