From the first post in this thread:
BUS SCHEDULE.
Animal Kingdom/Blizzard Beachonce per hour
Leaves Shades of Green on the half hour. Starts a half hour before park opening. (Usually at 0830)
On Extra Magic Hour days, Leaves Shades of Green starting at 0730
Leaves Animal Kingdom at 10 minutes before the hour (Space #7 in charter bus area)
Leaves BB on the hour (Space #1 in bus area, from 1000)
Hollywood Studios-once per hour
Leaves Shades of Green on the half hour (Starts a half hour before park opening)
On Extra Magic Hour days, Leaves Shades of Green starting at 0730
Leaves DHS on the hour (Space C6)
Typhoon Lagoon/Downtown Disney-once per hour
Leaves Shades of Green on the half hour (starting at 0830)
Leaves TL at 10 till the hour (Space #9 from 0950)
Leaves DD on the hour (at the top of lot E by the red building)
Last bus leaves DTD at midnight
Ticket and Transportation Center-every 20 minutes
To reach both Magic Kingdom and Epcot via monorail
First bus leaves Shades of Green a half hour before park opening
On Extra Magic Hour days, Leaves Shades of Green starting at 0730
Bus leaves TTC starting at 0840 (space #3)
For your first day, I'd say there is no way you're going to get into the buffet and eat and be out in time to get to MK at opening. Service is slow (you have to wait for your check etc) and sometimes the buffet is slow - wait for an omelette or waffles.
The Shady Shuttle will be running, but if it's out with another family you'll have to wait (or walk). Add about 15 min at least. Then you'll have to wait for the MK monorail. Monorail will be running plenty early enough for you - it needs to run early enough to accommodate the earliest ADRs .
To gt to Epcot for your early ADR - you didn't say what day, but if there happens to be an EMH at MK that morning the TTC bus will be running, then hop to the epcot monorail, which again, will be running. You can also try to walk to Poly/Shady and take the Epcot bus from Poly, but I find that actually takes longer, as you never know when that bus is going to arrive.