With your budget I would look at Morningside, Grange, Newington, Merchiston, Colinton, Murrayfield, Blackhall, Ravelston, Trinity, New Town. All lovely safe family areas.
Out of the city I'd look at Linlithgow to the west and North Berwick and Gullane to East.
Remember Scotland has offers over system and houses in Edinburgh have been selling for 20% over for last few years. But property market everywhere is slowing a bit.
If you grew up here you'll know about the peculiarity in Edinburgh with 25% of secondary children being in private schools. This means state secondaries don't have the same social mix as elsewhere. The "best" state high schools are considered to be Boroughmuir, Gillespies, Royal High, Craigmount.
Also Firhill,Trinity and Balerno get a good name.
If you get in catchment for any of them you really can't go wrong even if you do end up going for private.
The private day schools are all much of a muchness in terms of results but each have their own character so best to go to the open days or arrange to visit. Families often choose by location. Watsons or Heriots for south side and Edinburgh academy, ESMS for north. St. George's if you want single sex girls. There is also Clifton hall, a very small school just outside the city. The other schools all have a lot of boarders.
All the schools have children travelling in from all of the Lothians and Fife.
I'd advise having a serious think about secondary education before you settle for being too far out of the city. If you decide on private your DC may have over an hour travel each way. You will also be taking them in to Edinburgh at weekend for sports matches, parties, social life!