We will be there during your dates (4th-10th for us)! As someone else said, get the Unofficial Guide. I'd suggest going a step further and subscribing to the website operated by the same people, touringplans.com. Through the website, you can investigate every hotel room, park, attraction and diversion in Disney World. You can browse their touring plans to learn how to tour each park efficiently, and even make your own personalized touring plans and find out, based on the crowd level prediction for each day, the best order to tour in order to avoid waiting in line. It's cheap, too!
With 6 days, here's the plan I would suggest. (It's what we did last time with 6 days for DH, myself, our 6YO son and 4YO daughter, and it worked very well, no hoppers required.) Of course, it depends in part on when you arrive/depart. On all full days, plan to arrive in a park before rope drop (park opening). I also suggest bringing Pop Tarts, granola bars, fruit or cereal from home (you can buy milk at the resort) and having breakfast in the room before you head out each day. I'd suggest using snack credits for breakfast (muffins and such), but those credits tend to be more valuable mid-day when you are parched and overheated, for things like frozen Mickey bars and Dole Whip floats and citrus swirls (available at Aloha Isle in Adventureland, Magic Kingdom, and just as good as their reputation). Another tip: never pay for a beverage! Not only will you get them with every meal on the dining plan, but you can get free iced water from any CS location at any time. Just ask.
Day 1 (Monday)
Morning - travel/check-in (lunch en route or CS lunch at resort or MK)
Afternoon/Evening - tour Adventureland, Frontierland and Liberty Square in the Magic Kingdom. Have a CS dinner in the park. (I wouldn't plan a TS dinner the first day, just in case there's a travel delay -- WDW will by then be charging $10 a head for all ADR no-shows.)
Day 2 (Tuesday)
Morning: visit Hollywood Studios and tour most or all of what you want to see there. CS lunch in the park.
Afternoon: return to the resort for a 3-hour nap/swim break
Evening: Return to Hollywood Studios for a TS dinner (anywhere but Hollywood & Vine!), do whatever you missed in the morning, have a Fastpass+ reserved for Toy Story Midway Mania (which has the longest waits of any attraction at WDW) and see Fantasmic! (If Fantasmic is too late at night for the kids, skip it.)
Day 3 (Wednesday)
Morning: visit the Animal Kingdom and see everything you want to see there. CS lunch in the park.
Afternoon: return to the resort for a 3-hour nap/swim break
Evening: Visit Disney Springs (formerly known as "Downtown Disney") and have a TS dinner at Raglan Road (ask for a table in the room with the step-dancers and order the Serious Steak with Bread-and-Butter pudding for dessert -- you'll thank me!

) or another 1-credit TS of your choice. If you don't care to visit Disney Springs, visit a resort for a unique dinner experience (e.g., 'Ohana at the Polynesian, Hoop-de-Doo at Fort Wilderness, Whispering Canyon Cafe at Wilderness Lodge, any of the Animal Kingdom Lodge restaurants). If you still have energy to burn afterward, consider watching fireworks from the Polynesian, the Electric Water Pageant from Fort Wilderness/Wilderness Lodge, or simply watching the animals on the savannah at AKL.
Day 4 (Thursday)
Morning: visit Epcot and tour Future World. CS lunch in the park.
Afternoon: return to the resort for a 3-hour nap/swim break
Evening: return to Epcot and tour World Showcase. TS dinner at Biergarten, Tutto Italia or other restaurant of choice.
Day 5 (Friday)
Morning: visit the Magic Kingdom and tour Fantasyland. CS lunch in the park (will be out of pocket if you had 2 CS meals on Monday).
Afternoon: return to the resort for a 3-hour nap/swim break
Evening: return to the Magic Kingdom and tour Tomorrowland. Have a TS dinner at Crystal Palace, Liberty Tree Tavern or Be Our Guest. Stay late to watch the fireworks.
Day 6 (Saturday)
Morning: Pack up and check out. Leave your bags with bell services if you will be returning (e.g., to catch a Magical Express bus to the airport.) Visit your park of choice (it's always MK for us) to revisit each family member's 1 or 2 favorite attractions in that park until it's time to leave. If you didn't do any 2-credit TS dinners, you'll have one credit left. Use it for a nice brunch in or near the park you plan to visit. (For MK, good choices would be Crystal Palace, the 'Ohana character breakfast, Chef Mickey's, etc.) Otherwise, have a CS lunch out of pocket. Use up any remaining snack credits today also -- even if you just end up with rice krispie treats to take home.
NOTES: When we go in May, the only changes we'll be making to the above itinerary are: (1) we will be parkhopping, mainly for the purpose of dining in Epcot in the evenings, as some of the best restaurants are there; and (2) we're skipping the Animal Kingdom in favor of more time at the Magic Kingdom and an evening at Hoop-de-Doo Revue. Although the AK has terrific live shows, a nice safari ride and a beautiful and immersive theme, its major attractions are too rough for my family (e.g., Dinosaur, Primeval Whirl), too wet (Kali River Rapids) or have a height requirement that eliminates at least one child (48" - Expedition Everest). We've done the safari twice and have nice (nearly identical) pictures from both times -- plus, most of the same animals are in our local zoo -- so won't feel like we're missing much. For first-timers, though, I'd recommend it.
TO ANSWER YOUR QUESTIONS:
- Snack-shipping: absolutely you can. Some folks ship their clothes and toiletries ahead, too!
- Best Parades: Epcot has none and I believe the HS parade has been discontinued (somebody correct me if I'm wrong). The MK 3pm parade is good. The AK parade we have never seen (we've always been done-and-gone by 1pm).
- Best Character Meals: For princesses, your best 1-credit deal is the Akershus princess meal at Epcot's Norway pavilion. For Mickey and friends, there are many options, but the one I hear friends talk about most often is Chef Mickey's at the Contemporary Resort. For Pooh characters, there's the Crystal Palace, and for Lilo and Stitch (ugh), breakfast at 'Ohana at the Polynesian Resort.
- Great Live Shows: In Epcot, there is terrific live entertainment all around World Showcase. The Chinese acrobats (see your times guide for that day for details) are really good. The Animal Kingdom has 2 highly-regarded shows, Festival of the Lion King and Finding Nemo: The Musical. In HS, my favorites are Beauty and the Beast, The Little Mermaid and the Indiana Jones Stunt Spectacular.