Source for all these: Official Site, Tickets & Passes http://www.atac.roma.it/page.asp?p=229 Click the links/labels for information on this page, but the URL does not change everytime, so I could not link to each section directy.
Almost all of the Rome can be traversed by foot for tourism purposes, with an occasional one-use transport ticket EUR 1.50, but it needs careful planning, as Machines do not dispense any change mote that 4 EUR, you insert a note for which the machine isn't able to dispense change, then that note will be spat out by the machine (for more see comment by joernano in comments), or are out-of-work (mostly).
Also there is a time-limit on these tickets(100 Minutes). It means that you are allowed to change as many metros or stops as long as you stay inside the turnstiles, & unlimited bus/tram transport, as long as all this is within 100 minutes of Stamp Validation.
Careful, as any time you used up for waiting the metro @ Platform will also be counted in this as many stations have Validation Stamp m/cs at station entrance, & platforms are either on other level, or not-in-direct-sight. The fines for travelling with no-ticket/expired-ticket/non-validated ticket is Minimum EUR 100, which will be reduced to EUR 50 if paid within 5 days.
Valid for100 minutes from first validation; just for a single journey on metro/train urban route, even using different lines, but without passing through the turnstiles.
The most convenient/no-nonsense option will be CIS WEEKLY INTEGRATED TICKET a 7 Calendar Days weekly Ticket, valid from the date of first validation, to the 23:59:59 of 7th Calendar day, for the price of EUR 24.
Valid until midnight of the seventh day including the day of validation, for unlimited journeys within the city of Rome.
Also, 72 Hours / Three Day Transport Ticket (not Museums/Attraction Pass) is not EUR 36, but only EUR 18, valid for only/all public transport within the Rome City (Some Circle, last stations are marked by red on Official Transport Map http://www.atac.roma.it/files/doc.asp?r=385
72 Hour Ticket - valid for unlimited metro, bus, and train travel within Rome for 72 hours from validation. € 18.00
There are packs of 10 Tickets at discounted price (Offline/Paper based, multiple tickets can be used at same time, if multiple passengers. Online version is tied to one person, & can not be shared.
All these above works with any kind of Public Transport (in Rome City only, both Airports not covered). Not all tickets are available online, but all are available on Machines , Tobacco Shops.
I will also be in Rome for 9 days coming month.
Various Maps at same website http://www.atac.roma.it/page.asp?p=18