Subway and buses can be used using the same tickets. Tickets can be bought at selling machines in each subway/train station and at selected bus stops using credit cards or cash, or at most newspaper kiosks and tobacconists.
Tickets of most interest for you are as follows:
If you have a mobile Internet connection available, you can download the official app “Muoversi a Roma” (also in English) from ATAC (iOS and Android applications), the local transportation company, to have timely information about bus/subway lines, timetables, and waiting times at each bus stop.
If you plan to visit museums, have a look at Roma Pass, which also includes transportation.
To call a taxi dial +39063570.
If you wish to pay by credit card, ask the operator a cab with a card reader. Price for an average jorney within the city will be 15-30 Eur.