Saya,
For such a short time frame, it would be possible to make a quick trip to London, Paris & Rome, allowing 3 days or so in each.
As you're "first time travelers to Europe", I'd suggest pre-reading the Guidebook Europe Through The Back Door. That will provide you with lots of information to "travel well" (including "securing our belongings", and hopefully avoid too many problems.
Starting in London is a good idea, as there will be an easier introduction to the European culture (aside from the aspect of "two peoples separated by a common language"). You shouldn't have any trouble finding flights to either LHR or LGW.
After London, take the EuroStar to Paris for a few days. Paris is a wonderful city and well worth having a look at.
Finally, use one of the budget airlines from either Paris CDG or ORY to Rome. If you book early enough, the prices are often VERY cheap (however these increase as the flight fills). NOTE their Terms & Conditions carefully, as tickets are often non-changeable and non-refundable once booked! You'll need to allow at least 4-6 hours for that, and be sure to PACK LIGHT as they have very tight luggage weights.
After touring in Rome, fly home from Rome / FCO. With such a short trip, using open jaw flights is absolutely the best idea. You shouldn't need a rail pass, so you won't have to budget for that.
Don't be too concerned about "looking like tourists". The locals can always tell, regardless of how you dress.
You may also want to have a look at the country or city-specific Guidebooks for the places you'll be visiting, to get ideas on places you might want to see, opening & closing times, costs, Hotel & Restaurant info, etc.
Don't try to see everything in one visit. As Rick says "assume you will return".
Happy travels!