Ooh I love giving Edinburgh advice
As previously mentioned, Morningside is brilliant for families, there is also, at that end of the price range, Barnton, Cramond, the Grange, New Town (which is where Stockbridge is), Dean Village and Duddingston.
Stockbridge, Cramond, Dean Village and Morningside are like little villages, they have a very villagey feel, and are packed with lovely little independent shops, restaurants, bars and have good schools and parks and things.
Cannonmills is right beside Stockbridge in the New Town and is just as nice, but property is slightly cheaper.
Barnton, the Grange and Duddingston are a bit more suburby, though Edinburgh is very small and nowhere is more than 15 minutes or so on a bus. And the buses are great, frequent, fast, reliable and comfortable.
If you are looking at a lower budget, there is Willowbrae, Currie, Silverknowes, Leith, Holyrood/Abbeyhill, Portobello, Liberton.. all nice places, and all still have some very expensive properties in them, however there are much more lower priced properties in these areas.
If you really want to live in the centre, then try anywhere in the Old Town; Royal Mile, Canongait, Cowgate, Grassmarket, Lauriston, Marchmont, or anywhere around the Bridges (though the bridges tends to be full of students).
Hope this helps