We know we probably won't be able to do too much in that span of time
Actually, you should be able to do a LOT in 6 days. The big thing to avoid on April 1st (Easter Sunday) are the big National sites, like the Colosseum, which are free on the first Sunday of every month. That might be a great day to do a walking tour or two, and most guidebooks - including Rick Steves - have some of those mapped out for self-touring; no guide needed.
The Vatican Museums will be closed Easter Sunday and Monday (April 2) but you should pre-order tickets for a day that they're open to avoid standing in a long ticket line, You will still need to pass through security but that line moves pretty quickly. The page with their tickets and tours is here:
http://www.museivaticani.va/content/museivaticani/en/visita-i-musei/scegli-la-visita/visitatori-singoli.html
"Open tour of the Vatican Museums and Sistine Chapel" is a general-entry ticket for self-touring and not a guided tour. However, you may want to look at the 3-hour combo tour (guided) of museums + St Peter's for accessing the church directly via the tours-only back passage.
For the Colosseum, Forum and Palatine, (one ticket covers all three), pre-order here:
https://www.coopculture.it/en/colosseo-e-shop.cfm
"Open ticket" is the choice for self-touring the main section of the arena. I am guessing that the underground and Belvedere tours are long sold out for the week you'll be there, although you could take a look and see if tickets are still available. I personally don't think either of those areas are must-dos but other folks have enjoyed them.
Those are really the two places with the longest lines so you shouldn't stand around long to get into anything else. If you want to see the Borghese (excellent museum), reservations are mandatory, and it may be booked up for that week but give a look:
http://www.tosc.it/tickets.htm?affiliate=T2C&sort_by=event_datum&sort_direction=asc&fun=erdetail&doc=erdetaila&erid=912678&language=en
A day trip to Orvieto by train might be up your alley, or to the ruins at Ostia Antica.
A guidebook is a necessity for picking and choosing what might interest you in Rome so buy one (and check your local library for others) as quickly as possible and spend some quality time with it.