Here is my plan on the paper:

Could you please let me know if it id doable.

Any changes/recommendations are welcome.
Please ignore numbers, i just used them to mark attractions on my map.

Day 1-2:

Colosseum (1)
Roman Forum (6)
Circus Maximus(4)
Capitoline Hill and museum(7)
Foro Trajano(14)
Bath of Caracalla (11)

Day 3:

Vatican muesums (9)
St.Peter Basilica(5)
Castel Sant'Angelo(10)

Day 4:

Trevi Fountain (2)
Spanish Steps (13)
Pantheon (3)
Piazza Navona (8)
Piazza Colonna (15)
Piazza Popolo(12)

You have one glaring omission to me---The Galleria Borghese
You need to reserve a 2 hour time slot, but not to be missed. It would be my # 3 top attraction in Rome.

bobthenavigator, actually I do have it, but forgot to include in the list above, not sure what exact day I can make it.

Day 3 is very light - unless you're gping to be exploring and walkking throught a lot more areas. the first 2 days are way too busy - unless yuo plan on just looking briefly at the outside of things versus actually visiting and seeing something.

If you get a copy of the Michelin guide from the library it will tel you how long you should allot to visit specific major sights.

The first half of your Day 1-2 can be done in a day (or even an afternoon); we did and we were very thorough through the Palentine/Roman Forum.

If you have flexibility, I'd do the Galleria Borghese on the day you go to the Piazza del Popolo, since the Villa Borghese is adjacent to that Piazza. Try to get a reservation as early in the day as you can stand; the waiting area can get pretty zooey! You can then wander the gardens back to the Piazza del Popolo, get some lunch, walk to Spanish Steps then Trevi Fountain, Colonna, Pantheon to Piazza Navona...

You also could do Piazza Navona on your Day 3: the Castel Sant'Angelo isn't far away. Or go explore around Campo dei Fiori then...

What days of the week? Some things are closed on certain days.

Wed, Thurs, Sat and Sun.

Stay away from The Vatican on Sunday! People flock to St. Peter's Square each Sunday to see the Pope give a blessing from one of the windows of his Papal apartment. (And they arrive before the blessing and take in St. Peter's Basilica. And especially if it's the last Sunday of the month, the Vatican Museum, too.)

Piazza Colonna won't take much time at all.

BC

I was there for five days last month, and I think this is a pretty good list.

I doubt that the days will work out in the order you plan them to, you can do all this stuff in the time allowed, at least on the surface.

Two things I would suggest you think about: (1) what do you do if it rains? I would concentrate on museums and churches and limit the amount of just walking around looking, but you may have a different response. (2) Do not underestimate the physical challenges, even if you are in good shape. In particular, the forum is very rough underfoot, challenging ankles and calves, and the city is in fact built on seven hills, and some of them are steep. If you can't keep up the pace you have set for yourself, don't beat yourself over the head.

I disagree with BC on staying away from the Sunday Blessing. Whenever I have people traveling with me (usually first timers to Italy), I always take them to the Sunday Blessing. Most remark it's one of the highlights of their trip. I would not attempt the Vatican Museums on the last Sunday though.

Why are your days split? Are you doing a daytrip on Friday? If so, I would change the daytrip and plan the days this way.

day 1 - Vatican area
day 2 daytrip ?
days 3,4,5 your other sites - buy the Roma Pass (good for three consecutive days), book the Borghese for day 3 to get free entrance from the pass. Use your other free entrance for the Colosseum/Forum/Palantine on day four putting the Capitoline after the Forum (use the pass for discount at Capitoline). The pass free entrances is the first two sites you visit with Borghese and Forum area the most expensive it's best to use it for them first.
http://www.romapass.it/p.aspx?l=en&tid=2

nytraveler, I was going to suggest that day 3 is too HEAVY!

Seriously, I would break it into two parts (hopefully the OP's hotel is close by, like in the Campo dei Fiori or Piazza Navona area).

Spend the morning seeing the Castle St. Angelo, the Vatican museums and other parts of Vatican City. Then take a long relaxing lunch break nearby, or find some other way to get off their feet for at least two hours (maybe even a nap?). Then return and see St. Peters - allowing at least two hours in the the basilica - allow an additional hour if they plan to ascend the dome (GREAT views, but if they are not totally worn out by then, they WILL be after that climb.

Somewhat depending on where your hotel is located, you could visit the following on the way to or from other activities: Trevi Fountain, Spanish Steps, Pantheon, Piazza Navona. We visited Piazza Navona every day to sit at a cafe and people-watch, and went to the Trevi and the Pantheon almost as often.

My hotel is near Trevi Fountain

Agreed that a visit to the Galleria Borghese is from all reports a must (wasn't able to visit it when I went to Rome). It would fit easily into your "Day 1-2" or "Day 4." You need to get a timed entry ticket for this attraction. At present, your "Day 1-2" shouldn't take more than a day and a half. And your Day 4 is very light -- you should have time to see other things here.

I don't think "Day 3" is too much, myself. Fortunately, you can always play this by ear, and if you find the Vatican Museum has (perhaps understandably) taken up more time than you thought, you can always go to the Castel Sant'Angelo on "Day 4." I personally was very underwhelmed by the Castel, but not everyone shares this opinion.

Depending on how much time you have, I'd also consider seeing the Etruscan Museum (it's in the same park as the Galleria Borghese and you should be able to combine it with a visit here) and the St. Peter in Chains Church for its justly famous Michelangelo sculpture of Moses.