Wow- you got some appallingly bad advice in the first 3 posts. the later ones are much better. #10's post could have been written by me: excellent advice.
I have a 2 year old and we spend about 6 weeks a year in Italy, near Rome, with her. We also lived in Rome for 2 years before she was born so I know the city well. My daughter made her first trip to Rome when she was 6 week sold. She's actually named after a Roman Saint...
anyway, to address some of the issues:
Rome is NOT unsafe. It is as safe as any other major European city, keep your normal travel wits about you and you'll be fine.
At this age, the key to baby's happiness is happy parents. So if you want to go to Rome, go. And don't leave the baby behind- you're a family now and this is your life together. It will be different, but you'll enjoy it just as much. And if one or both of you are working, I cannot imagine you not wanting to spend your free time with your child
However, a few things DO change when you travel with a baby. You'll find that you'll be travelling at half or a third of the speed you used to. Why? baby needs a nap, needs to be fed, needs a diaper, etc. This will set the pace. Therefore, accept you won;t see all. And for that same reason, tours are not a good idea, as somebody else will be in control of the pace. YOU need to be in control of the pace.
In 2 1/2 years of travel we've never brought along a stroller. she's 15kg now, and still travels on our back. we use an Ergo baby carrier which is in my experience the best piece of equipment. Flexible, light, and with the right support. At 6 months, they're so light that a baby carrier is really the best way to travel. And kids love it- being close to mommie and daddy and being able to see from high up is their idea of fun. (You should see my daughter come run when i put on the Ergo) (the reason I and many others don't like Baby Bjorn is that many of their models don't provide lumbar support- meaning that all the weight rest on your shoulders. With the Ergo it rests on your hips)
Our daughter has done quite a few museums in her Ergo- no problem.
If you can, breastfeed. That gets rid of all the other hassle related to food. Always clean, always safe, always at hand. We also co-slept when travelling till she was 8 months old- again, simplifies life. If you don;t co-sleep, you will need to think about a baby bed.
Consider getting an apartment. If you stay in town for a week , it's nice to have more space and be able to make some simple meals (or clean bottles etc). Also, at some point they need a schedule and darkness to sleep. And if you're in a hotel room, what do you do when baby sleeps at 8PM? maybe she's a solid sleeper (ours isn't) but if you have a separate bedroom she can sleep there while you still have a normal dinner etc in the living room. Many apartments can provide baby beds, but you will need to ask beforehand, or bring your own.
We do go out with her (and again, at 6 months its easy) but once they're mobile it becomes harder and an apartment is really much better than a hotel room. We have good experience with these people: http://www.cross-pollinate.com/
It's already been mentioned, but Italians love children. Expect lots of attention. Other parents will encourage their kids to say 'ciao' to yours, and you'll be complimented continuously, as well as being quizzed about its sexe, age, and name. Don;t be upset if people touch- they're just friendly. On a number of occasions we had our daughter taken out of our hands in restaurants and being carried around by the staff so that we could enjoy our meal.. be open and friendly and you'll get plenty of good vibes in return.
Oh, i just read in your post #8 you're from Canada... it's a long flight. but gain, travelling with a 5-6 months old child is a breeze compared to 1 year old. Make sure to ask for a bassinet when booking. Hopefully your baby will sleep- Our daughter did he first trans-Atlantic flight when she was 10 weeks old- that was the easy bit. At the age of 2, it's just 9 hours of hard work... So perhaps you'd better go to Europe now and save MOntreal for when she's a toddler... ;-)