There are two routes you may like to consider. US-1, which south of Palm Beach can get busy and A1A which runs between the intercoastal canal and the Atlantic Ocean. The latter is an incredibly scenic route with access to the coastal beaches in places. It is of course, much slower than the routes quoted by Buck, which are super highways without anything much interesting along the way.
It is quite easy to swap between US-1 and A1A, in fact, in places it is essential, as A1A is not continuous. All along these routes fishing and boating is the main activity, and you will find many great restaurants which specialise in seafood.
Ft Pierce is a nice place to stop. The Arcade Building on US1 with its murals is a highlight. The town also has many other murals.
Before you get to Cape Canarveral A1A heads back to US-1, over a bridge. It is that very bridge where the crowds would watch the launch of the space shuttles. I was fortunate to be there for a launch in August 2009.
Enjoy
Lifey