I'm from Miami, but live in Texas. If you're staying on South Beach proper, you can rent some bikes from Miami Beach Bicycle Center (601 5th St, Miami Beach, FL 33139, (305) 531-4161). I know they lead an escorted ride once a month, but I haven't done that ride in 5 years, at least.
Another shop that rents bikes is Elite Cycling & Fitness (13108 S Dixie Hwy, Miami, FL 33156, (786) 242-3733). They're definitely "off the beaten path" relative to South Beach, but I think they're a better shop. The owner is a former US Olympian, and he knows his stuff.
Finally, try Mack Cycle & Fitness (5995 Sunset Dr, Miami, FL 33143, (305) 661-8363). They're the "gold standard" of cycling in Miami. They sponsor many of the century rides, but I'm not sure if they rent bikes.
As for rides, there are definitely a bunch of good rides you can do. Depending on your speed level, you can try Everglades Bicycle Club (evergladesbc.com). They have three different Saturday rides, two of which are leader-led, with varying distances and speeds. Their Sunday ride is great. Two groups: 17-18 and 22+, leaving from Miami City Hall and going to Hollywood. It's exactly 50 miles, round trip. The rides leave at 7:30A on both days.
Other rides you can try are listed here: http://www.themiamibikescene.com/p/group-rides.html
Have a good time on Miami Beach. Remember, LeBron doesn't play there. He plays in Miami.