Its been some time since I was in the area of A1A, but the last time I headed towards that direction I turned around. The stretch in Broward county was just too treacherous for me, particularly around Hollywood/Dania -- narrow high-speed roads. It does get a bit better once you begin to enter the city of Aventura.
Beyond that you can cycle all the way down past Art Deco hotels to South Point Park in South Beach and walk your bike out onto the Pier to muse as the cruise ships leaving port. You can cross the 3 mile causeway west to Miami and take the Underline (bike) path to South Miami and the over to Old Cutler Rd or go a couple blocks east over to Brickell Ave.
From there you can either take another (roadies) causeway to Miami-Dade's other popular beaches (less tourist more natives) all the way to Florida State park (see avatar). Or continue south through Coconut Grove to Old Cutler and all points in south Dade. Any of these choices are fine and popular cyclist routes.