I would go the other direction, for two reasons -- driving time, and real beach. (Incidentally, we just did BOTH this week.)
In Florida, you have two kinds of beaches -- Atlantic coast with broader beaches and real ocean waves, and Gulf coast with a lake-like beach with no waves. Both beaches are very nice, but the Atlantic side is the real ocean beach. To me...Florida Boy.
I would go to Cocoa Beach. Google Maps tells me 49 miles/48 minutes, but I'd plan on an hour. Road is great, no construction or congestion. It IS a toll road, however.
From the airport, you just take FL 528 which is an expressway the entire way to Cape Canaveral. When you get off the expressway, you will be on Florida's legendary A1A. Continue south and just take any street with a president's name east (left) to the ocean. You will find free public parking and free public beach access -- AND pretty uncrowded beaches. If there is no shower there, go another block or two until you find a public bathroom and shower. If you go further south into Cocoa Beach, you will find more congestion and paid parking.
UNLESS...
you go to Ron Jon Surf Shop (just south of FL 520 and A1A -- about 4 1/2 miles from the end of the xway). Park FREE in Ron Jon's parking lot and walk a block to the beach. Ron Jon's itself is well worth a stop for the obligatory classic Florida teeshirt and/or every imaginable kind of beach accessory. (we spent 3 hours shopping there on Tuesday!)
Personally, I would go to Ron Jon for the more built-up, touristy beach atmosphere which is, well...um, unlike Wyoming (one of my favorite states).
The other option -- is to go west from Orlando (I-4 to I-275) to somewhere along St Pete Beach (Gulf coast).
Google Maps says 113 miles/1 hour 55 minutes to Reddington Beach (one of the closest Gulf beaches). But there is a lot of construction in Tampa (I drove this route twice last week), so I'd say a little more time and some significant stress if you're unfamiliar with Tampa.
When you get there (Reddington or anywhere else along the Gulf Coast), you will find a beautiful, tranquil beach with waves measured in inches, not feet. Very beautiful and peaceful.