Travel with 3 year old and 1 year old... stroller question

I would opt for the umbrella stroller. I think you will find it easier when going up
or down stairs and also on trains.

I would opt for two umbrella strollers or one and a back pack carrier, theres no way a child at three will not need a stroller, and obviuosly the baby will need one or you will be carrying him/her all the time.
Yes a double stroller will be a pain. The streets can be rough( but not all cobble stone, and even many cobble stone streets have at least set stone walkway/sidwalks, just some are really narrow )and I had a double stroller myself, ( had three kids under 6 at once) and those things are never that easy to manevour in crowds at the best of times even here( Canada and States) with our pavement and nice ramped curbs.. There are some good strollers that have little steps on them that the older child could at least stand/sit on part of the time for rests( purpose built that way) it will be a a balancing act finding one thats light enough and portable enough but I would look. Frankly when our first two were under 3 we kept travel simple, one destination places like Hawaii or cruises so I can't say I would choose Europe for tots, but many people do and have a great time,, extra work and planning but it can be fun ( apparently lol ) . Even with a car there will lots of walking, parking is not usually right beside tourist hot spots, so you will likely find in some places walking a fair bit .

We've been here in Europe with little ones, traveled all over. You need a sturdy umbrella stroller. A cheapy one will get the wheels broken off in about 5 minutes on the cobblestones. The double stroller will be difficult for the narrow sidewalks and some of the narrow doorways. A backpack carrier is awesome. We used ours with our first until about 4yrs of age. Depending on size of the child, of course. If there's two adults, then two backpack carriers would be great to get through some of the museums....just keep passing cookies back and you've got an hour to see it. There's another good option that I don't know how available it is in the states currently. Lots of strollers here have a step/platform on the back where an older child can stand/kneel/sit. That would be an excellent choice. Lastly, look at the carseat regulations, if they're doing random border checks they could also make sure the kiddos have the correct carseats.

With two young kids, and lots of transportation going on, you are going to need flexible options. Don't bring a double stroller. You will find it more of a liability than an asset. I would recommend a Peg Perego Pliko + a carrier that you are comfortable with the 1yo in. The Pliko has "steps" on the back that your 3yo will be able to stand on when (s)he is tired. When the 3yo is asleep on their feet, you can put the 1yo in the carrier. Or, you can leave the Pliko in the hotel/apartment and take a carrier when a stroller isn't practical. The Pliko isn't the best for cobblestone streets, but it's better than a lightweight umbrella stroller. There's a reason why those aren't sold here in Europe. Will you have a car the entire time? You'll need carseats, and carrying those - along with the stroller, kids and luggage could really get to be cumbersome if you take a train or even just in the airport. You will definitely need to think about how much you can physically manage. You may be able to rent carseats, but I wouldn't recommend it as you don't know whether the carseat has been in an accident before, and there's no guarantee it won't be filthy or have been recalled. Good luck with the planning.