We just did DisneyWorld this past December-the week before Christmas. My DH had a conference on site for work (lucky!) This was our second time at DisneyWorld.
Because my DH is a federal employee-we have stayed at Shades of Green military resort on Disney property both times. It's VERY reasonable and clean and perfectly nice, but I absolutely hate the restaurants on site there and the service is terrible. Will not make that mistake again-regardless of room price!
In December-there are LOTS of extras at the park-special Christmas parades, etc-all cost extra and require advance purchase tickets. I think with 2 six year olds who can probably make a longer day-I would spring for at least 1-2 of these things. We didn't because a 2 year old couldn't make it through the long day.
I highly recommend staying on resort property so you can ride the monorail and not have to drive in and park everyday and have the flexibility of going back to the room to rest. Get a room with a kitchenette or at least fridge and microwave so you can keeps drinks and snacks and small meals in the room. Park food is super pricey! Don't buy souvenirs in the park-order or shop at Disney store first (particularly if you know what your kiddos will want) and bring along as a surprise-way cheaper!)
Plan, over-plan in fact-the more organized you are about what you are doing, reservations for special meals, etc-the better the experience is. Use fast passes all that you can-start early in the morning and if you stay on Disney property-take advantage of extra magic hours to get in early or stay late. It's cheaper to stay longer, if you can-do at least 4-6 days, particularly if you plan to do multiple parks. Magic Kingdom is still the best part of Disney in my opinion and we always spend the most time there. Epcot had some great food and picture sites and a few fun rides, but overall can be a littled dated and probably not worth an entire day. The Animal Kingdom is a lot of fun for kids (basically like a deluxe zoo) The Hollywood themed park is more appealing to older kids, teens-wilder rides (might be too wild for 6 year olds) Time might be better spent at Universal if you are looking for the whole movie experience with younger kids. Bring rain ponchos-there's always a downpour! Bring a backpack or something to carry with you in the park-Florida weather changes like crazy and you might need to change clothes, have snacks, etc. In 3 days at the park in December-we experienced 30 degree temps with wind and upper 70 degree temps with rain...you never know!
Have a blast! And know in advance, you will likely spend more than you want-even with good planning. It's just a fact of Disney!