Once we got past the 4 or 5 "top sights" that a weeklong visitor will hit, we started finding museums without the lines and crowds. In fact it's quite common for us lately to be the only people in galleries, one corridor after another filled with antiquities. Just outside we have to box our way down the sidewalks to get to these places or out to lunch.
The best deal in Rome that I know of is the Museo Nazionale Roma, which is spread out over 4 locations. The 7euro ticket is good for multiple entry in all of the sites over 3 days. (It's worth the price for the bathroom access alone.) The Palazzo Massimo location has the most amazing collection of frescoes and mosaics we've seen, and the best curation and presentation, also.
Across the street is the Terme (Baths) di Doicleziano. Much less remains than of the baths of Caracalla, but it's easier to get to, and the attached museum is a good place for the scholarly impulse with in-depth presentations on proto-history of the region and on epigraphic history.
Palazzo Altemps has some excellent overlooked sculpture and gives a good insight to how antiquities collecting happened in one family.
And Crypta Balbi show how one layer of history is piled on another and how archaeologists work to peel it back. We've seen the same point made elsewhere, but this was the most expansive presentation of it and it's at an interesting location, on a road that was important to the city in Caesar's time and important today. (We ride the bus through it every day.)