The apartment you booked is on Via Giulia... one of the most BEAUTIFUL streets in Rome... I would NOT waste time at Termini. Grab your bags and enjoy a drink in one of the beautiful cafès there!

Another possibility, why *not* check your bags for a few hours and visit some of the sights in the vicinity of Termini?

Within a few minutes walk lie the basilicas Santa Maria Maggiore, Santa Prudenzia and Santa Prassede. Even closer, you'll find the Chiesa di Santa Maria della Vittoria, the Basilica di Santa Maria degli Angeli and the rest of the baths of Diocletion, and the Museo Nazionale Romano di Palazzo Massimo. These are all major sights -- if you've planned to visit even one you could save considerable time later.

Termini is not a "pleasant" place to hang out, I'd pass through there as quickly as possible. *BUT* if you're willing, check your bags there and walk over to St Maria Vittoria (three blocks) and see Bernini's St Teresa in Ecstasy - it could be a great start to your Rome trip.
On the other hand, the areas you're staying in is great for hanging out - Pza Farnese has great people watching..
If you want to go to the Colosseum on your own, early, download a free podcast (Rick Steves is good) just go the the tourist office across from the Forum, get your RomaPass and walk by the queue waiting to get in at Colosseo. It includes entry into two Rome museums and a transport pass good for three days too - best deal ever, esp. for all those things you'll do w/o a guide. Includes discount to all but vatican museums too. After the Colosseo, you can also walk past the Forum line too... :)

Your apartment is in a great location as others have noted. I would not do a tour on day 1 but would explore the area. Via Giulia itself, Piazza Farense, Campo de' Fiori, Piazza Navona, Piazza della Rotunda--see as much or as little as you have time/energy for.

I also highly recommend visiting the Palatine as well as the Forum and Colosseum. I don't know how tickets work if you are on a tour but if you buy them yourself they are good for all three sites.

Thanks everyone. We have decided against booking the tour the day we arrive, although we may book the Passeggiata walk that first evening. My son thinks it would be a fun way to start our trip - and will ensure we don't go to bed too early. I am glad to hear positive things about going it alone at the Forum & Colosseo. Maybe we will just go without a guide to those sites and book the Context Underground Tour for later in the week as a special experience. That certainly would leave lots of flexibility in our timing. And it is so nice to hear such positive things about where we are staying! I have booked a car service from the airport and will hope to find a cafe in the neighborhood where we can hang out for an hour or so before meeting our landlord if need be. I can't wait!