Busses start running typically at 7am for Character Breakfasts.
If you have a Character Breakfast at the Contemporary, Polynesian, or Grand Floridian, take the bus to the Magic Kingdom, and the Resort Monorail from there (the MK Bus drops off at the Transportation Center before 7:30)
If you're having breakfast at the Magic Kingdom, keep in mind Busses and Monorails will not let anyone out at the Magic Kingdom until after 7:30. Busses will just be dropping off at TTC, Monorails will just zip right through the station.
If you are having breakfast at any other Disney resort, take the bus to a hub (Magic Kingdom, Animal Kingdom) and then transfer to the appropriate resort bus. Busses don't typically run from resort to resort, however if you're the only one on the bus,
sometimes the bus driver will take you directly there, but it depends on if their schedule allows it. Depending on where you're going for breakfast, you might want to ask the bus driver which hub to go to.
I would factor in 30mins for travel time... although you might get there in less. If you're a little late, not to worry. Most places will hold the priority seating for about a half hour, they understand about travel times and stuff.
Enjoy.