Walt Disney World covers 43 square miles. The closest 2 parks are Epcot and Disney MGM, which are separated by water, or basically just across Buena Vista Blvd from each other. The furtherest out is Animal Kingdom. From DTD, it is about a 10 -15 minute drive, if you catch all of the lights, and know the short cuts. There are basically 2 main roads in WDW, and from them, you can get anywhere you need to go. They are World Drive, which starts at 192 and ends up by the Magic Kingdom. From here are exits for all parks and resorts. The other is the aforementioned Buena Vista Blvd. This runs from rt 535 past DTD, and ends at the All Star Resorts. From here you can directly reach DTD, Typhoon Lagoon, Carribean Beach Resort, Boardwalk Resort, Swan and Dolphin, Disney MGM Studios, Fantasia Miniature Golf, Coronado Springs Resort, Blizzard Beach, Winter/Summerland Miniature Golf and the All Star Resorts. It also intersects roads leading to Epcot, Old Key West, Port Orleans Resorts, Pop Century, Wide World of Disney Sports Complex, Animal Kingdom, Magic Kingdom, and Animal Kingdom Lodge. Stop by Guest Relations, if you aren't staying onsite to get a map, or if you are staying in the Disney owned resorts, there will be a pamphlet that you will receive on check in and it has a fold out copy of the map. It looks confusing, but usually castmembers at Guest Relations or the front desks can give you the most direct route to get you where you are going.
As for walking, you can safely walk from Epcot to Disney-MGM Studios, following the path along the water. You may also walk from Boardwalk, Yacht and Beach Club and Swan and Dolphin to either of these parks and Fantasia Miniature Golf. You may also walk from the Contemporary to the Magic Kingdom. You can walk between the Polynesian and the Grand Floridian, too. You can also walk between Blizzard Beach and Winter/Summerland Miniature Golf. It is kind of a long walk, but there is a pathway that connects DTD to Saratoga Springs, as well. Walking between other parks and resorts is highly discouraged due to the heavy volume of traffic and lack of sidewalks.
Have a great time!