Hi Panayiota,
Congratulations on your upcoming wedding! You have chosen a magical place for your honeymoon.
Once you leave Disney property, you will either need to hire a cab (ask for names at your resort) or rent a car to get around the Orlando/Kissimmee area.
Renting a car will likely be your most cost-effective choice. In the past, our family has rented a car with National or Alamo car rentals. The National/Alamo Car Care Centre is close to the Magic Kingdom toll booth and their phone number is (407) 824-3470. It is open Monday to Friday 6 am to 8pm, and Saturday and Sunday 6 am to 6pm.
The bonus with using this location is that they currently have a courtesy shuttle pick up at the resorts. Just call (407) 824-3470 at least one hour (and up to 24 hours) before you need a pick up. You can also return your car at this location and hop on the courtesy shuttle back to your resort, as long as you return your vehicle at least 15 minutes prior to their closing time.
National/Alamo also has a second location at the Walt Disney World Dolphin Resort. Their phone number is (407)934-4930. It is open 8am to 4pm every day but does not have a complimentary shuttle service.
For places of interest outside of Walt Disney World, you can visit the websites of the Florida or Orlando Visitor Bureaus.
Congratulations once again!