Hi, Jo Ann! The Swan and Dolphin resorts are such great resorts! You really can't beat the location and the pool area is just awesome.
I always have a rental car when I am in Walt Disney World, but I have to say I still use Disney transportation at certain times, but because of your resort's location, I don't think you will be driving much. You will want to take the bus to Magic Kingdom. It is so convenient to have your resort bus drop you off right at the front gates. If you want to head to Epcot from there, you can simply hop on the express monorail at the Magic Kingdom and transfer at the Transportation and Ticket Center for your monorail to Epcot.
From the Dolphin, you can take the 15 minute walk over to Epcot or 20 minute walk to Hollywood Studios or there are boats that will transport you as well. With little ones in your group, I think the boats will be a better option for you.
So the answer to your question? Park your car and leave the driving to Disney! Remember that parking at the Dolphin is about $12 per day (prices are always changing) and it is a few dollars more if you plan to valet park.
I hope you have a great stay at the Dolphin, and although you didn't ask about dining, I highly recommend Piccabu that is right in your hotel! It is one of my favorite breakfast locations; make sure to try their potatoes, so yummy!
If you have any other questions,
let me know! Have a wonderful vacation.