Going to Rome. What method of buying tour tickets should i go with. Buying them when i am in room, through the hotel or through an online site like expedia, or through other sites?
Start by trying to book with the official sites 1st. Here's a few to get you started:
Colosseum: https://www.coopculture.it/en/ticket_office.cfm
Vatican
Museums: https://biglietteriamusei.vatican.va/musei/tickets/do
Borghese Gallery:
https://www.tosc.it/biglietti.htm?affiliate=T2C&sort_by=event_datum&sort_direction=asc&fun=erdetail&doc=erdetaila&erid=912678
If the days/times that you want are sold out then you may want to see if the tour operators (Walks of Italy, Context Tours, there are lots more) have availability.
I'm curious why you are waiting until you get to Rome? Tours/tickets through the official websites for some of these places can sell out in advance so there's an advantage for buying ahead if at all possible. If you're on a budget, tickets/tours through the official websites are almost always bargain priced as well.