1st trip to Rome and Florence

Advance booking is mandatory for the Borghese Gallery in Rome. If you want to see the underground area of the Coliseum, you should book that as well; some posters have been disappointed when they were unable to get tickets. Reports from recent travelers to Rome indicate that the Vatican Museums are unbelievably crowded. If that's something you plan to visit, you might consider an early-morning tour. I haven't done that, but the Walks of Italy Pristine Sistine tour gets positive comments here and gets you into the museums an hour before the regular opening time.

Not sure about Florence. For both cities, check the places on your must-see list for possible closure on Sunday or Monday as that will affect how you lay out your visit.

I imagine your lodgings will be fine. If you post the names and addresses, we probably have folks here who can comment on the specific neighborhoods.

Had you asked this question BEFORE buying the passes, I think most of us would have advised against doing so: they're not usually recommended for Italy. Some advice on this from Mr. Steves:

https://www.ricksteves.com/travel-tips/transportation/trains/italy-rail-passes

Will I be able to use the Eurail pass for the train from Rome airport
to Termini station?

It's my understanding the the Eurail pass is only good for the Leonardo Express if it's a first-class pass.

https://www.raileurope.com/european-trains/leonardo-express/how-to-book.html?gclid=COOUp4PfoM8CFQkMaQodJ7kG_g

Also, since our time in each city is limited what are the sites that
we definitely should buy tickets for ahead of time?

Rome:
Vatican Museums: http://mv.vatican.va/3_EN/pages/MV_Home.html

Possibly the Colosseum...although you can get around that by purchasing your tickets at the Forum or Palatine where lines are much shorter (1 ticket covers all three). Otherwise: http://www.coopculture.it/en/the-colosseum.cfm.
Pre-booking is a necessity for the Underground/third ring tours (same website).

Galleria Borghese: pre-reservations are MANDATORY for this one:

http://www.tosc.it/tickets.htm?affiliate=T2C&sort_by=event_datum&sort_direction=asc&fun=erdetail&doc=erdetaila&erid=912678&language=en

Florence:
Uffizi and Accademia: http://www.b-ticket.com/b-ticket/uffizi/default.aspx

We are renting an apartment in both cities with locations within 15 to
30 minutess walking distance from the rail station. Are we asking for
trouble?

What kind of trouble are you expecting...other than 30 minutes can be a long time to walk with luggage if you have a considerable amount of it? In that case, take a cab.

In addition to the previous comments on Rome, I suggest getting reservations for the Uffizi and Accademia museums if you plan to go to them. This will save you some time. Florence is much smaller than Rome and the historic center can be traversed end-to-end in about 20 minutes. Any lodging in this area is perfectly safe, including areas in Oltrarno across the river.

I question if you should have booked your train trip to Florence on your day of arrival. What if your flight is late?

There are numerous trains per hour to Florence, and none of them sell out seats, so there is no reason to buy in advance, unless you want to save money, but at the risk of losing everything if your flight doesn't land on time.

Also I'm not sure about the wisdom of buying the Eurail pass in your case. The full fare ticket from Rome to Florence is only €45, while the ticket from the Rome Fiumicino Airport to Roma Termini (Rome main station) is only €15. So €60 is all you need to get to Florence. To come back to Roma Termini you need another €45. I presume that on the morning of your flight home (presumably early in the AM), you may want to take a taxi to the airport (€48 flat fare) rather than the train, especially if your hotel is not walking distance from the Termini station.

If you are in Florence only 3 nights (2 days), you barely have time for one side trip. If you go to Siena, you wouldn't even take a train, you'd go by bus. If you go to Pisa and Lucca, the train fare to see both cities is under €20 (round trip).

Your Rome stay is also not long enough for more than one side trip. Let's say you go to Orvieto, the round trip train fare is €15.50. If instead you want to go a bit farther to Naples and/or Pompeii, then you need €44 each way (full fare on a high speed train), plus a few euro for the commuter train to Pompeii (where your Eurail pass wouldn't work anyhow).

So once you add everything up, your singularly purchased total train tickets per person would amount to, at most, about €140, or €215 if you go to Naples instead of Orvieto (US$160 and US$250 respectively). I hope your Eurail pass was cheaper than that, because otherwise you didn't make a wise investment.

For museum bookings follow the advice in the above comments.

Thank you for your reply as it was very helpful. Our eurail pass was cheaper that the individual prices of the train trips that we had planned that is why we opted to go that way. Also, on the day of our arrival in Rome we will arrive at the airport by 13:00 . We booked a train to Florence at 17:20 allowing for flight delays and long custom lines. If we have a longer delay than that we have back-up plans. I appreciate your advice.

Thank you for the recommendations for which sights to book ahead of time. I am looking at the Walks of Italy tours since they were recommended by several travelers who have posted here as well as Rick Steves. Our Eurail pass is 1st class so according to the responses here it should be good for the Leonardo express from the airport in Rome to Termini station. Wondering which side trips as worth it for such a short period of time - has anyone ever taken a wine tasting tour from Florence into Tuscany, in particular Chianti?

Marylee, to add to Roberto's comments, I'd echo his thoughts that with just 2 full days in Florence and 3.5 in Rome, you don't really have much time for side trips. There is at least a couple day's worth of things to see in Florence (we've spent 4.5 days without running out) and 3.5 days will just barely scratch the surface of Rome. I guess it depends on your to-do list and what you're interested in but IMHO, 1 side trip from Rome - and I personally wouldn't recommend that for a first-timer - would be the max. Great cities, both, with a lot to keep you occupied! :O)

At 15:08 there is a high speed train that departs directly from the Fiumicino Aeroporto station and goes to Venice (stopping in Florence along the way). If you can make it, and your booking allows changes, you could take that one and avoid changing trains at Roma Termini, you just stay on board of the same train (which still stops at Roma Termini).

May I ask you how much your Eurail pass was? I thought Eurail passes (even Italy only passes) were at least $160 for just a 3 day use in a month.