The most important thing is schools, adn in particular high school catchment areas. The way Edinburgh works is there tends to be a very good high school next to a neighbouring not so good high school. So, for example, on the north side, Trinity and the Royal High are very good schools, and sandwiched between is Craigroyston, a terrible school. This also happens on the south side. Liberton is a better school than Gracemount, but both are relatively poor compared to James Gillespies, close to the city, and Firhill, nearer to the outskirts of the city. The pick is probably Boroughmuir, also close to the city on the south side, but with a strange, elongated catchment area, going all the way to the border of Edinburgh. Next to it is Tynecastle, a poor high school.
So a good idea, as a starting point, is to go onto Edinburgh City Council's website and download the school catchment areas. Also check that the primary school , if your children are of Primary age, is in that High School,s catchment area.
There are some subtleties. Trinity Academy is a better school (north side,) than Broughton High School but, Broughton is home to the national academy of drama and dance, neither of which are offered at Trinity. And a few other things to keep in mind. Most of the good schools are full, so if you have children who will be inserting into a class somewhere in primary or secondary, you would need to write to the schools to request a place.
I live in Edinburgh, on the north side because I like to live next to the sea. I prefer Glasgow as a place to live (the usual thing in Britain, the further west you go, the warmer the people) but the great thing about Edinburgh is the weather. It doesn't rain that much and has some of the highest sunshine hours in Britain, after Brighton.
Something to keep in mind - snow. Lovely to look at, lovely to see, but not outside my front door please. I really wouldn't recommend living above the snow line. Balerno and Juniper Green and Currie have lovely houses, possibly in your price budget, but just keep in mind, the last bad snow season (2010) Balerno was literally snowed in for 6 weeks. That means trekking, through deep snow, to the entrance of the estate (a large estate) and catching a bus from the main road, because the council doesn't clear the side roads, just the main road. But hey, if you prefer winters that look like real winters, then areas above the snow line I would recommend are the catchment areas for Balerno, Currie and Fir Hill high schools. Don't get this wrong though,. Especially for the balerno, Currie, Juniper Green areas, because the neighbouring high school is Wester Hailes, one of the worst in Britain.
Ig you are catholic, make sure you are in St Thomas Aquinas catchment area. the other two catholic schools are terrible. If you like the west side, the Royal High is better than Craigmount High School and both are better than Forrester. On the East side, make sure if you live there, you are in Portobello catchment area. Same as for Currie, the neighbouring high school, Castlebrae is the worst in Scotland.
it's really important to be within the right catchment area. You can apply to go to a neighbouring school, but put it this way, at our school, Trinity, no people requesting an out of catchment place at Trinity were given a place. At our local primary school they have had to build 4 new classrooms this summer just to cope with demand. Same at Wardie, the other local primary.
On the housing side, the problem with all these schools is that the house prices in their catchment areas are probably out of your range. Maybe an ex council house up in the Fir Hill area. But that means living in Oxgangs. Yes, you get a house, and are in the catchment for a great school, but the area....
Will your children be wanting to learn Mandarin? Edinburgh Chinese schoool is based at Drummond Community high. Don't whatever you do, live in that catchment area. Again, it is a terrible school with massive drug problems. Hardly any of their kids stay on for highers. Contrast that with Trinity, which isn't the best of the best but has a retention rate for highers of 82%.
I know this can be tedious, but it's worth going onto google maps, type in, say, Balerno, go onto street view and have a wander around. Get a feel for the areas. Portobello, - if you live that side, make sure you live well away from Seafield Road. That's where the sewer works are and in summer there can be a right pong.