Transport Info - going from Jeddah to Makkah, medina

First for Ehram: there are certain zones for Ehram that you shouldn't pass them without wearing Ehram, they are called Mawaqeet. If you are coming by plane, you should wear Ehram before boarding.  Intention of Omra can be made once flight crew will tell you that the plane is passing Ehram zone. You and your family should make intention at that stage and say: Labayka Allahumma Omra.

For transportation: there are mini buses from airport to Mecca at 200SAR for the whole trip.
I suggest that you rent a car from airport and drive down to Mecca.  It is easy from Jeddah airport to Mecca and you don't need to go inside Jeddah.
From Mecca to Madina if you rented a car; just go back to Jeddah and the highway itself will take you to Madina without any need to enter inside Jeddah as well.

There is a stand in Jeddah called 'Baab Makkah', where you will find plenty of cars/vans/busses ready to go Makkah for Sr10-15 per seat.

for Madinah, i prefer a bus service called saptco, which charges Sr60 per side and the journey is very comfortable since the journey is 5-6 hours long