If you like history, you should probably plan nearly a full day at the ancient Roman sites, and if you're interested in the Vatican, nearly a full day there.
There is a single ticket to the Colosseum/Roman Forum/Palatine Hill site, for 11 euros, and it's good for 2 days. I would suggest going as early as possible in the morning, because it heats up later in the day and there isn't much shade, especially in the Roman Forum. You could start at the Colosseum, before the crowds arrive. Then, I would visit the Palatine Hill, walking toward the Roman Forum through the Palatine Hill. You will see the huge remains of the imperial palace, and many other ancient buildings. There is a little museum on the Palatine Hill that has a very interesting exhibit on the lower level about the development of prehistoric Rome. When you've finished seeing the Palatine Hill, you can walk right into the Roman Forum without exiting the site.
At the opposite end of the Roman Forum from the Palatine Hill, the Capitoline Museums is up a set of steps, on Piazza del Campidoglio, designed by Michelangelo. This would be a good place to spend the hot part of the day. I could spend hours in this museum, which has a really great collection of ancient sculpture. On the lower level, there is a passageway to connect the two museums, and there you can also see the foundations of the ancient temple of Jupiter. There is a good cafeteria inside the Capitoline Museums, with a view over the Roman Forum. There are also other overlooks that give you great views of the Forum.
When you're finished there, you'll probably be too tired to do much else. You might want to go to your hotel for a rest and change of clothes before dinner.
When you go to the Vatican, I would start with St. Peter's Basilica, because the Vatican Museums tend to have long lines in the morning. You can get an audio guide for the Basilica, which will help you understand what you're seeing. You should aim to hit the Vatican Museums around lunch time (1 PM maybe). There is a cafeteria inside the Museums, which may be a better bet than the restaurants in the vicinity, which are often mediocre and overpriced. You can also get audio guides for the Vatican Museums. If you want a tour, you can see this page, which has the official Vatican tours:
…vatican.va/3_EN/…MV_Info_Servizi.html
This page has information about hours and ticket prices:
…vatican.va/2_IT/…MV_Info_Orario.html
After the Vatican, you might want to visit Castel Sant'Angelo, which is not far away, on your way back to the center. On the upper level there is a bar on a terrace which has great views of Rome.
On one of your other days, you might want to go to Ostia Antica, which is one of my favorite historical sites:
http://www.ostia-antica.org/
You should also spend part of a day wandering around the historic center. There you can see Piazza Navona,
One way to keep costs down would be to get a Roma Pass, which gives you free admission to two sites, and discounts on others. It also gives you a separate 3-day transit pass, for unlimited rides on buses, trams, metro, and a few urban trains. It costs 23 euros, but it can easily pay for itself; the Colosseum/Roman Forum/Palatine Hill ticket itself is 11 euros. It doesn't cover the Vatican Museums.