Another welcome to the forum, Sandra-
To add to Laurel's excellent advice....
If you are staying 3 nights in Rome, you more accurately have 2 FULL days plus some odd hours to work with. If the 19th is arrival day on a flight from the US (or similar distance away), the odd hours you're left with may be a jet-lagged fog. As well, flight-related delays or cancellations are not uncommon so for both reasons, you don't want to book any tours or pre-purchase museum tickets for that day.
The historic center of Rome is reasonably compact and there's usually something interesting to see en route from one place to another. We've been in the city for as long as 4 nights/3.5 days without taking any public transport at all, other than from the airport! It may depend on your sightseeing schedule and stamina but I'll recommend doing as much on foot as possible; Rome is a great walking city! Do be aware that the metro station closest to the Spanish Steps (SPAGNA) will be closed to renovation during your visit, Laurel has provided the public transit website so spend some time with that?
I don't recommend a multi-day pass for the amount of time you'll be in Rome. You can't use them to get from the airport, and you may not get the good of one over just two days. That said, if you want one just for the convenience then there's a 48-hour pass that you only have to validate the first time you use it, and works for the buses, metro and urban trains.
https://www.atac.roma.it/en/tickets-and-passes
An on/off bus for 3 days.... I think you meant a hop on-hop off bus? Those have been widely discouraged as good means of transport in Rome. There are just too many places they can't go, and there have been many reports of dirty buses with surely staff and broken equipment. Reports as well of riders being stuck at pick-up points for long periods of time due to to-capacity buses. not having room for additional passengers.
Tickets:
The Vatican Museums (the Sistine is within the museums) website is https://www.museivaticani.va/content/museivaticani/en.html.
You are late in the game for booking one of their tours but you could try. If they're completely booked, let us know and we'll suggest some independent companies which offer tours, albeit for higher prices.
Colosseum/forum/Palatine: official ticketing site is https://ticketing.colosseo.it/en/
(For some odd reason it appears to be down this morning; must have some kind of tech issue.)