The simple answer? You can only do it by picking some and choosing to save the rest for a future trip. Try and do too much and you'll just end up frustrated, hot, exhausted, and possibly coming away hating Rome.
I'll emphasize that last one as people who report that they really disliked the city often gave it too little time and spent all of what little they had at the Top 10, fighting the mob. The mob can get real old, real fast if you don't balance it with some breathing room.
Spending time with a guidebook is an excellent suggestion. Pick some things you want to see and group your sightseeing by area so that you're not running willynilly from one end of the city to another. Then prioritize the chosen attractions according to YOUR interests. Make good use of the evenings for outdoor things, like Trevi, Piazza Navona, etc. so that you can use your daytime hours for attractions with visiting hours.
That said, I'll be the dissenting vote for the Vatican Museums. Along with the Colosseum, it's the most-visited attraction in Rome, and doing that + the basilica would put you squarely in the middle of another mob fest on your 2nd of only two full days. It can also eat a LOT of time. If you have an interest in art, I'll second the vote for the Borghese because it has excellent crowd control, a superb collection, the building itself is REALLY interesting, and you will see it in the two hours each group is allotted. As already noted, the park it's in (Villa Borghese) is nice for a wander and some breathing room. Walk all the way through to Piazza del Popolo, maybe stopping at one of several outdoor cafes for a rest and a beverage. Santa Maria del Popolo, on that piazza, is well worth a stop, IMHO. Anyway, I'd save the Vatican for another trip unless it's tippytop of your list and something YOU feel you must do. Not everyone does.
LOL, my husband, who has a healthy regard for things architectural and artistic, was less blown away by the Vatican basilica and museums than he was by the Borghese, Basilica of St John Lateran and some other churches in Rome. So there you have it. :O)
But the Borghese can be skipped as well in favor of just walking and looking, popping into a couple of cool, dark churches along the way or a bar seat for rehydrating and some people watching. It's up to YOU how you wish to spend your time, and you simply can't do it all so don't even try.
piazza's such as Piazza Michelangelo
You probably mean Piazza del Campidoglio, which the master designed. Do that on the same day as the Colosseum/Forum/Palatine as it's in that same area.