The Roman Forum & Palatine Hill are both situated in the same archeological area and admission to both is covered in your ticket for the Colosseum. The tickets are valid for 48 hours, making it easy to split up these attractions into separate days. Or you can simply spend one day transitioning from one attraction to another as they are all located in the same area. I'd advise The Colosseum on one day and The Roman Forum & Palatine Hill the next. The latter has so much area to cover, and you don't want to be rushing to see it all. Plus this will keep you from standing in line for one or the other, as you can get there early both days.
As I usually advise, get there early. It is in my opinion the best way to see these attractions, while avoiding long lines. I'd also suggest that if you plan to visit many of the city's archeological sites and museums, all while taking public transit, grab the Roma Pass and save yourself some money. You can get a 2 or 3 day pass, which covers public transit and 2 free entries (on the 3 day pass) into the sites of your choice, plus discounts on all others. The Roma Pass excludes The Vatican & St Peter's. There are also other passes and tours you can choose to purchase while in Rome, all designed to streamline your experience, but you have to weigh the costs against what you think you'll actually see.
We had booked some before hours tours (not covered by passes) and had plans to be super early to the other attractions, so we opted to just purchase a 7 day public transit only pass (the main stations sell them at kiosks), which was essential for us as we stayed outside the city center. This made getting around Rome super simple and was one less thing we had to stress over. I'd highly recommend getting a transit pass which I believe come in 24hr, 3 day, and 7 day increments.
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