I'm spending a couple of weeks in Rome in July, and I'm thinking about how to schedule the things I want to do. I appreciate that this is the high season, and everywhere will be busy, but are weekends noticeably worse? I.e. should I spend weekends pursuing more off-the-beaten-track sights to avoid larger crowds at main attractions, or does it not make that much of a difference? Also are any particular activities / locations worse on particular days?
I've been to Rome several times for long durations and I did not notice any particular day of the week to be more busy than any other. However, weekdays I found were less busy and especially in the early morning you can see tourist sights and not be as overwhelmed with other bodies. Since you will be there in the summer the sun rises early so if you don't mind waking up early, you'll be able to avoid some of the bigger crowds. Also, late at night (maybe ~9:00+) the city does tend to quiet down and sites like the Pantheon are beautiful to sit down and drink a glass of wine beside. The locals tend to venture out more at night so you get a better sense of the real Rome.
Regarding your question about things being worse/better on particular days, I'd say that the Vatican is quite busy on Sunday due to Mass. Also, try to avoid the major tourist sites within Rome when it's really busy (mid day - evening) because seeing the Trevi fountain overrun with tourists really takes a lot away from it's beauty and can ruin your experience. Inside St.Peters really early morning is also a great time to visit because it's fairly quiet and the sun comes through the front glass windows and can create some gorgeous light beams inside the cathedral.