There is a so-called "Stadtbahn" which stops at the Burgau car park and the market place and drives around the town. The tour takes about 40 minutes and you'll get a few information on the town, in German, English, French and Dutch. However there was really limited information, I had excepted more for that price (adults 5,-€, students 3,50 €, children 2,- €). And especially the French and Dutch speakers had bad luck as in most cases we had already passed the place that just was talked about. The cobbled streets made it rather uncomforable in that "train", and we drove exactly the same streets that we already had walked the morning - there are not so much streets!
The only advantage was that the train drove up to the castle which we hadn't yet done because that's not so simple with a pram and wheelchair. At the castle you get 5 minutes to exit the train and enjoy the view, but that's not much time. The train from there goes down to the Burgau parking, I regret we didn't stop the tour there and walked down through the old town, that would have been nicer!
But at least my nephews enjoyed the tour, and would have loved to make a second round - and probably also a third one!
There are four large car parks near the old town and a good parking guidance system which shows you where you can find a car park and the number of free spaces. We used the car-park Burgau, which offers space for over 300 vehicles and is very convenient when approaching the town from the south. It's just a few minutes to the old town from there. A day ticket costs 4,- €, I don't recall how much it was for an hour.
In the old town you'll find very limited parking spaces and there is an access control system that restricts access to the town to Monschau Card holders (residents, merchants etc.). You can get a "short visit ticket" at the access control machines, which is rather for hotel guests to (un)load their cars and emergency than for "normal" visitors.
There are lots of ways to get to Monschau. Several busses come there. However, you can buy a regional ticket from the train systems that includes Aachen, Eupen, Liege, Maastricht and a few other cities between Germany, Belgium and the Netherlands. The bus from Eupen to Monschau runs every 2 hours from the train station and is part of this ticket!