In case you are staying for a while and want to drive around on your own, you could look for a car to rent.This way you are not limited to the public transportation routes and times.
You could also travel outside of the city and explore the rest of Hungary; however, you will always have to fight for a place to park.
Here is a list of car rentals operating in Budapest:
They offer a variety of different types of cars and additional services, such as child seats, gps, Wi-fi hotspots, extended warranties and adding extra drivers.
Furthermore, you can choose the location where you want to pick up and drop your car off.
Check their websites for more details on rules and conditions, because they may vary form company to company.