There is a relatively straight drive from Miami to Orlando. When you leave the port head to I-95 and go north about 20 miles to I-595. Go west on I-595 for about 3 miles to the turnpike. Get on the turnpike and go north. You will go all the way to Orlando on the turnpike. The cost is about $12. Head north past Kissimmee to Osceola parkway. Exit at the Osceola parkway and you will stay on this until you reach Disney. (cost on the Osceola is about $2) The turnpike is a limited access road but does have food , bathrooms and fuel about every 35 miles or so. Fuel is a bit high so make sure you are filled up before you get on the turnpike. Food depends on the rest stop, but they have subway, burger king and nathans at most of them. Cost of food is a bit higher than off the turnpike, but definately easier than exiting, looking for someplace to eat and returning to the turnpike. It should take about 4 hours from the port to Disney property. This estimate is based on normal traffic (not during rush hours) and 1 fifteen minute bathroom stop.
The Osceola is a pay highway but is not limited access, you will think you are on a regular city street, but to my family it is the fastest way to get to Disney.
There are other ways to get to the turnpike from I-95, but the one I have given you is the easiest and does not take you on city streets with traffic lights and possible wrong turns and missed signage. Until you reach West Palm Beach on the turnpike traffic will be heavy but usually moves rapidly. Just stick close to the speed limits and you should be okay.
Have a great trip!