Welcome to the boards,
you are in the right place for help.
First things first, take a deep breath. You're going to Disney!!!! And with your whole family, that is amazing. I'm sure it seems overwhelming now, but you have plenty of time to book a trip of a lifetime.
I'm going to ask a few questions before offering up any advice.
Food/Dining
What is your dining style? Do you eat three big meals a day? I ask because this will help me determine if I would recommend a dining plan, and what type. I am not a big breakfast eater at all, so I am usually content with a bagel from a quick service in whatever hotel I happen to be staying at, but do like to sit down for lunch and dinner.
A few observations based on the age of people in your party.
1) I see that there are a few small children going, so I'm going to assume you would want to do a few Character Meals (these are meals, usually some sort of buffet or limited menu options, where while you eat characters come around and visit the table). If this appeals to you, I'd suggest figuring out what characters the kids would like to see. Basically, the options are Princesses, the Fab Five (Mickey, Minnie, Donald, Goofy and either Pluto or Daisy), Playhouse Disney and Winnie the Pooh. There is also a Lilo and Stitch breakfast. Once you know what characters they would want to dine with, you can determine which, if any, character meals would be a good fit for the family.
2) With the small children, if you aren't planning to return to the resorts for a break in the afternoon, I would suggest planning a sit down meal for lunch. These are referred to as Table Service. This gives you a few benefits. First, no one has to worry about finding a table, gathering all the food, etc. Second, you are able to just sit and be served in a nice, air conditioned environment. While its not a nap, it is an excellent "rest" from the parks. Third, for the most part, the food options and quality are a bit better. Fourth, you're going over March Break, when Disney tends to be a bit more...crowded
Having dining reservations made in advance for a sit down meal means that the little ones are eating at a time that they are used to, and fed children are happy children.
Parks
I see you mentioned Universal. Just wanted to make sure you know that this is not on Disney property, so you would either need to rent a car to get you there and back, or book a shuttle. Given how many of you there are, it will likely be more economical to rent a car (or two). Also this will require a separate ticket. Maybe check out the various rides available there and make sure there are enough things for the younger ones to do to warrant the whole group going.
As for how many days in each Disney park...that's a hard one. As I'm sure you know, Disney World is comprised of four parks and two water parks.
Given the age of the children, your main focus will probably be the Magic Kingdom. I would think one day planned at each other park would be fine, although again, just pull up park maps on any site and you can determine what sort of rides your family would be most likely to enjoy.
The water parks at Disney are amazing (I am partial to Typhoon Lagoon), but make sure you check the average weather for when you're going to make sure everyone will want to be running around in a bathing suit. I went once on March Break and it was freezing. As a previous poster mentioned, water parks are an additional cost per person.
Ticket Options
The way it works is you start with a base ticket. This gives you access to the four main parks, one park a day.
You can add a "hopper" which allows you to go to one park in the morning, and then another in the afternoon or evening. You could, in theory, visit all four parks in the same day. The fee to add this is per person, and is an additional 57 dollars.
You can also add the "water parks and more", giving you access to the two water parks on property. The fee to add this is per person, and is also an additional 57 dollars.
IF YOU ADD BOTH the hopper and water parks, the fee is only 79 dollars per person.
I personally never hop parks. I wake up, I go to a park, I stay there. Or I nap in the afternoon and then go back to that park (as re-entry to the same park is allowed). I just don't like to spend my time travelling between parks, but that is just me. You will know better what your family would want to do.
Resorts
Disney has three levels of resorts. Value, Moderate and Deluxe. As you go up in level, you go up in size of room, types of amenities, etc. I have stayed at all levels, and actually at most of the resorts. The choice to stay at what level is going to be heavily influenced by two things. Your personal preference and your budget. A Value resort will run you on average 100/night per room. A Moderate about 200 a night and a Deluxe about 350/400 a night. Some are more, some are less. So my advice to you is to look at each resort and see what appeals to you most.
A few things to think about in this category.
Value's have double beds, most moderates and all deluxe's have queens. The decor is a bit more "in your face" and may appeal most to the children. I haven't seen it in person yet, but the pictures of the Art of Animation look super amazing. The only transportation option for Value's is the bus system, so it tends to be quite efficient but can also be quite busy at times.
Moderates tend to be "sprawling" in their size. Most have multiple bus stops within each resort, which can mean a bit of a wait at times. And a fair bit of walking to get to the main building from your specific room. Moderates are also mostly bus based for their transportation, with the exception of the boat to Downtown Disney from Port Orleans Riverside and French Quarter.
Deluxe's, while more compact in size than a Moderate, do tend to "share" buses. So the bus to MK from Beach Club will also stop at Yacht Club, Boardwalk and on occasion the Swan and Dolphin. But, if you are planning to be mostly at MK, I'd suggest staying closest to that park (meaning a Deluxe resort). I've stayed at the WL, the Poly, the Contemporary and the Grand Floridian, which are the four closest to MK. The latter three are literally on a monorail line, a phenomenal bonus when frequenting the MK, especially if there are little ones looking to go down for a nap mid afternoon. Also, you can take a monorail to Epcot as well.
Of the three resorts on the monorail, I love the Poly the most. Great theming, the restaurants are good, the pool is great, etc. Again, personal preference will factor in huge here. I just liked the resort best overall, and you could walk to the Ticket and Transportation Centre to catch the Epcot monorail, rather than take one monorail there, disembark, and then move to a second monorail. However, you can walk to MK from the Contemporary in about 5 minutes, so that may play into your plans as well. As andrencau mentioned, the Monorail is usually a big hit with kids (and adults!) so even though it took longer to get to the park riding the monorail, we usually did that instead of walking anyhow lol.
I realize this was a TON of information in one post, I'm sorry, I just love the planning phase of trips
Any specific questions you may have about anything, ask away!!!
for your upcoming trip
We will walk you through this and hold your hand and wish we were going with you! 
I have people call me for advice all the time, and they tell me everything they've bought, but haven't actually read. It's broken down into managable sections with specific sections devoted to specific age groups; that sounds important based on your description of who you're traveling with. That said--you'll have a great time, but a little advance planning will help alleviate you feeling overwhelmed. And cheer up--the last time I went, I took 327 people with me (literally a marching band), and everything turned out great
Yours will too!
