We used to live out on the Space Coast, in the Melbourne area. Absolutely gorgeous beaches, and lots of other fun stuff in the area to do as well. If you want more info, just let me know.
From Kissimmee, get on the 192 (one of the main east-west highways in Kissimmee) and head east. You will stay on the 192 all the way out, eventually it will cross all the way over to the coast. Most of the beaches there are free and have very good public access. From the beaches, you can get on the A1A and head north to Cocoa (Kennedy Space Center) and Daytona Beach, and be sure to check out what may be one of the biggest RonJon surf shops *I* have ever been in! LOL
The drive from Kissimmee out to the beach will take about 1 1/2 hours, depending on Traffic.