My tuppence worth would be to take this route. NEPN to Roseburn and then follow the newly Signposted Route 9 to Maybury.
Pretty much as Stickman recommends but instead of staying on the Pinkhill Railway to Paddockholme and Corstophine High Street, you take Meadowhouse Road and Dovecot Road to Broomhall.
This is my route into town/NEPN, the only amendment that I make to this is if my eldest is cycling independently in which case we'll take the Pinkhill Railway all the way to Jenners Depository and use the Pelican (soon to be Toucan) crossing of Balgreen Road and then Baird Drive and the WoL path to Roseburn. This avoids 2 unprotected road crossings and some minor roads.
The problem with doing anything North of Queensferry Road is getting across it. My routes tend to revolve around the 2 more easily accessible ones (to the West of Roseburn path)
The first is at the North end of Clermiston Road North, which has a Toucan to crossing and gives access through the Park to Davidsons Mains (and the NEPN). This does require ascending/descending Clermiston Drive, so fails the flat test.
The second is as per cb, at the foot of Drum Brae North, there is set of lights to cross QR and a cut through to get to Barnton Grove. You can get to Barnton Grove by continuing through the woods to the North of QR from the Clermiston toucan crossing.
The path as cb describes is very pleasant and whilst there is a small hill, it avoids the worse of Drumbrae. Unfortunately it is illegal to cycle on these paths currently and there are 6 steps to negotiate as you cross Craigmount HS, although I'm currently campaigning to change both of these. See previous thread.
You can avoid the steps if you take Craigs Road west to East Craigs Rigg and use the cut through at the bottom of ECR to access the pavement alongside Maybury Road to access Maybury.
Crossing QR further West of Drum Brae North is trickier as you have the choice of Barnton Junction itself (not recommended for a newbie) or the 2 stage pelican crossing past Cammo Road. Neither of which are particularly well connected with through routes and have narrow pavements.
Hope this helps,