Best way to maximize time is the same as at any city: sightsee by area. Get a map before you go and mark the attractions which interest you so you aren't jumping all over the place. Florence is compact so it doesn't take a lot of time to get from one to another; Rome is also very walkable but it can be a bit of a distance from, say, the Colosseum to the Vatican.
Pre-ordering tickets to eliminate long, long lines is also a must. It makes sense to do the Colosseum, Forum and Palatine together on the same day as they're very close to each other, and one ticket covers all three. Do St Peter's and the Vatican Museums on the same day, and pre-order tickets to the museums. For Florence, you'll want to pre-order for the Uffizi and Accademia if those are on the Wish List.
You also need to be aware of which days/hours attractions are closed (i.e. most Sundays for the Vatican Museums) and try to avoid free museum days if your trip should happen to fall on any of those. The first Sunday of every month is free admission to all Italian State museums…which in theory sounds like a good deal except that the biggies (Uffizi, Colosseum, etc.) will be insane. I'd advise choosing different activities on these days.
It's going to be hot and sticky in August: booking hotels with air conditioning is very definitely a good idea!
Lastly? Read. Have your college-age young people read. Use guidebooks and the net to come up with things that you all want to do and/or are willing to do separately on different schedules. For instance, your college students may be night owls and not early risers? Some of you may also have art museums high on the list where others would rather do ANYTHING but one of those! :O)