I have 3 young children and went a couple of times when 2 were in strollers. Like sympatico said, you have to fold up the stroller in order to board the parking lot trams so that takes some teamwork (one person holds the kids while the other wrestles the stroller onto the tram) but it wasn't that big of a deal. From what I remember, in order to park close enough to avoid the parking lot trams you need to arrive at the parking lot 10-15 minutes before opening time.
As for the drive to/from the parks and the traffic, it can be a little confusing & hectic the first time you make the drive from where you stay to the parks. But once you do it the first time and learn the route, it's not a big deal. Please don't let it stress you out too much. My wife is a big worrier when it comes to driving (she doesn't like to drive in places she isn't familiar with), and she is able to handle driving to/from a couple of times each trip.