I live in Edinburgh.
If you are coming up the A703 from Peebles, a lot of cyclists turn right at the Leadburn junction and take the A6094 through Auchendinny to avoid Penicuick, which is worth avoiding. (There's a steep climb in Auchendinny, but I still prefer that route.) Once back on the A703 the other side of Auchendinny I just stick to the A703 all the way to Hillend. As madcow says, the A703/A702 junction is unpleasant, but it is unpleasant if you are driving too.
(Last Saturday's ride, as an example of this route
https://www.strava.com/activities/5717196341 )
The question is how to get from Fairmilehead, where you cross the bypass, to the cycle path near Haymarket Station. The area outside the station is afflicted by tram tracks and much traffic and you don't need to go there anyway. My problem in trying to think about this is that I live fairly near the start of the cycle track so I'm trying to remember that you don't need all the wriggles that are in the map in my head!
Here's my suggestion:
https://goo.gl/maps/LVHWtdxPUjEGyQpZ9 heading for the Forth Bridge, I recommend you use the North Edinburgh Path Network Sustrans route. This uses old railway lines until you get out of Edinburgh at Crammond Brig, then a better-then-it-used-to-be path next to the dual carriageway until you peel off through Dalmeny. The signed route then wriggles less than ideally through some new housing but can be followed to the Bridge. Alternatively stick to the road all the way down into South Queensferry itself - this is nice because you get to cycle under the rail bridge and get the iconic view of the Bridges but has the disadvantage that you loose more height than you need to.
Hope that helps.