A couple more comments if weighing DIY versus a tour?
Hotels: I do recommend them over apartments for first-timers as the added support from a guest desk can be comforting on a maiden voyage. They also aren't any tougher to search than accommodations in U.S. cities you've never been to.
Use a booking site - I like www.booking.com - and search by date and desired price.
Don't let the number of stars throw you: those are more often based on amount of amenities an accommodation provides- such as room service or on-site restaurants - versus quality. If you don't need a lot of extras, then a 2 or 3-star hotel can be just fine if....
...it has decent reviews. Check reviews from previous guests, and check them on multiple sites: booking.com, Venere, Trip Advisor, etc. Ignore the glowing few which sound like advertising, the few simply too terrible to be believed, and those from tourists who are (obviously) ridiculously picky. The truth usually lies in the middle. You can also browse the forum for prior suggestions from trusty RS respondents to the hotel question.
Check their locations CAREFULLY. While there aren't really any "bad" areas in the centers of all 3 cities you list, some hotels that list their location as, say, Rome, can in actuality be way out on the fringes. That's not where you want to be if planning to hit the star attractions. If you find a few which suit budget but you're still confused about location, just post the question and you'll get help!
Read the fine print. Depending on the site you use, you may see two prices: one for non-refundable rate and another for free cancellation up to a certain date. This is common on booking.com. Those lower, non-refundable rates can look pretty attractive until or unless you have to cancel a trip. "Non-refundable", means exactly that: you're on the hook. Taxes are almost always included (nice!) but there is usually a small, separate per-day, per guest city fee which is not. Those vary depending on the per-night price of the hotel, and are usually asked to be paid in full, in cash, upon arrival.
Some folks like to book directly with the hotels they've chosen, and many offer have their own websites for doing so. For the ones which may not, a booking site can be nice as you can reserve your rooms at any time of the day - no time-zone difference - and there's no language barrier, should you be a little nervous about that. Again, though, you'll find that most hotels have desk employees who speak excellent to entirely passable English.
Time of year will dictate some desired amenities. For instance, air-conditioning isn't a given in smaller properties. If that's important to you - considering the current heat wave in Italy, I'd DEFINITELY want good AC if booking Rome in early Sept. - then read carefully. Same goes for in-room wifi.
All-in-all, not so different than finding a place to sleep right here in the States!