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I'm a big fan of the Disney Transportation system and use it extensively on every trip, but it is HUGE and can be intimidating if you don't know how to use it properly.  I'll go over the best ways I've found to get around the Disney World Resort using this free mode of transportation.

Monorail

There are 3 monorail lines that all meet at the Ticketing and Transportation Center, this is the parking area for the Magic Kingdom.

Magic Kingdom Express Monorail
This is the most direct route to The Magic Kingdom and just runs directly back and forth from there and the TTC.  There are only those two stops, so no getting lost :)

Magic Kingdom Resort Loop
This line goes from the TTC and makes a stop at each of the Magic Kingdom Resorts (Polynesian, Grand Floridian and Contemporary) while looping around to the Magic Kingdom entrance.  When getting on the Monorail at the Magic Kingdom entrance, only use this line if you are staying at one of the resorts or want the scenic route to the TTC.

Epcot Monorail
This line runs directly from the TTC to Epcot and back with no stops in between. So, like the express monorail line, there is little chance of getting lost here.  It's a nice long ride that takes you into Epcot for a bit giving you a unique view before pulling into the station in front of the entrance.

Going between Epcot and Magic Kingdom
The only way to do this is to transfer at the TTC.  You'll have to get off, walk down to the Monorail entrance area and then go up to the resort line you need next.  It's still a very fast and easy way to get between them.

Bus System
There is at least one depot at each Disney Resort where you can catch a ride to all four Theme Parks, both water parks, Downtown Disney, ESPN Wild World of Sports, etc.  Buses begin running about two hours prior to park opening so even if you have a breakfast before park opening, the buses are still a great option for getting there.

Every Theme Park has buses that will get you to any resort, so if you need to get from one resort to another(say for a dining reservation)...it's often best to bus to a park and then transfer there.

Park Exit Strategy
Using the bus system to get back to your resort at park closing can be a long wait since everyone and their brother rush to the exit as soon as the fireworks are over.  If you leave with them you WILL wait for an extra 30 minutes or more for a bus.

Keep in mind that the shops stay open for at least an hour after the park closing time, so we've found it's best to hold off on any shopping until this time and let the bus lines thin out.  This gets us the benefit of a shorter wait but also gets us precious time during the day that we don't spend shopping.

Boats
I'm not much of an expert on using the boats, but will tell you what I do know.

Epcot Area Boats
There's a boat line that runs from Epcot to Disney Studios with boats arriving every 20 minutes.  If you need to get between these two parks, want access to Disney's Boardwalk, or one of the Epcot resorts then this is a great option for you!

Magic Kingdom Boats
These boats run all over the Seven Seas Lagoon which is a man made lake next to Magic Kingdom.  From the TTC you can take a Ferry over to Magic Kingdom instead of the Monorail.  There are also smaller boats that will take you to the nearby resorts as well.  We used them to get from Magic Kingdom to Hoop Dee Doo Musical Revue and they worked really well.


Resort Hotel to the Parks

If you decide to stay off property, then you'll likely be driving to the resort each day or using a provided shuttle.  Staying at a Disney Resort gets you a parking pass for the duration of your trip good at all Disney locations that would normally charge ($14 a day adds up fast otherwise).

Getting to Epcot
If you are staying at a monorail resort (Polynesian, Grand Floridian or Contemporary) you can take it to the Ticketing and Transportation Center stop and change trains over to the Epcot Monorail.  This is the fastest way to get there and one of the key benefits to staying at these locations.

If you are staying at an Epcot area resort (Swan & Dolphin, Yacht or Beach Club, etc) you can either walk a short distance to the International Gateway entrance or catch one of the boats which will drop you there as well.

The fastest way from most other resorts is to drive and park at Epcot.  Getting there early will get you a spot up close, if you happen to book through AAA you'll get a special parking pass that gets you right up close anytime.

Last, but certainly not least, is the bus system which we use for most of our transportation on property.  The buses arrive every 20 minutes and will get you to Epcot in 15-25ish minutes depending on which resort you are staying.

Getting to Magic Kingdom
Monorail resorts get you to Magic Kingdom without having to even transfer to another monorail and are by far the fastest way, taking only a few minutes.  They also have walking paths if you feel up to it or are traveling before or after park hours.  Monorails usually run an hour before and after hours.

Driving to Magic Kingdom should only be done if you have to...it is the most time consuming way to get there.  You have to park, get across the enormous lot to the Ticketing and Transportation center and still take a ferry or monorail over to the entrance to the park.  Huge time sink :(

For most resorts, the bus system is by far the best way to get there.  The buses run every 20 mins, drop you off right near the entrance to the park and take about 15-25 mins depending on which resort you are staying.

Getting to Disney Studios
There is no monorail access to Disney Studios, but you can catch a boat from any of the Epcot Resorts that will get you there in relatively short time.

If you are staying at one of the Value resorts and have a car...driving to Disney Studios is likely the fastest bet.  You can be there in under 10 minutes.  If you don't have a car, the bus system will get you there in the normal 20 minutes from just about any resort on property.

Getting to Animal Kingdom
There is also no monorail access to Animal Kingdom and just like Disney Studios is a short drive from any Value Resort.  The buses will get you there in the normal 20 minutes or so and do so pretty consistently

IMPORTANT: If you have a breakfast reservation at Tusker House before park opening you should be leery of driving and parking at Animal Kingdom.  The parking lot sometimes doesn't open until closer to park opening time and you'll miss your reservation entirely!  We've always taken the bus from our resort and gotten there in plenty of time :)