From Disney Swan to Magic Kindgom
 

We are staying at the Swan for our first Disney trip. We will have our car (and two toddlers to shuttle around, with their double stroller). We have reservations at the Contemporary the same day we will be in MK. Would it be best to take Disney transportation from the hotel or to drive our car to the Contemporary and park? Also, is it true the buses from the Swan bypass the TTC. Thanks!!!

I assume you are asking about the bus route and if you might have to change between the Swan and the Contemporary and I cannot answer that (we usually stay at the Yacht and Beach nearby your own hotel). I would assume you could drive over and self-park without having to worry about paying to do so.

You can walk from the Swan to Disney Studios and then take their transportation to Magic Kingdom (this will be a bus). The other option is to take the boat from the Swan to Epcot and then take the monorail to the Contemporary/Magic Kingdom

Not sure on parking at the contemporary - they might limit it to registered guests since so many other folks can access the hotel using disney transportation

There is a bus to the Magic Kingdom right at the Swan/Dolphin.

A "bus to the Magic Kingdom" is not, as you probably know, going to be as convenient as parking AT the Contemporary. BUT what you could do is take the bus to the MK and then take the short walk to the MK monorail station (since you are, I assume, going to have the children in tow) and ride into the Contemporary. That would eliminate the need to drive, any worries about parking at the Contemporary, etc.