You could explore on Google maps for parks and some playparks are on Tripadvisor, but the local knowledge does make a difference.
Saughton Park is a real humdinger of a play park (also rose gardens and what-nots) and is between Gogar and city centre . The immediate area is probably not aspirational but I have a couple of relatives in Chesser/Murrayfield, one with kids and they are happy enough, and close to that park, so think safe and family and friendly but possibly a bit down-market. Saughton Park is well worth a walk or bus ride.
Public transport is very good in Edinburgh, explore www.lothianbuses.com. That website also has good maps, so you can spot green spaces. If you are working in Gogar you might be able to use the tram to get to work. It's packed during peak time but if you buy a ridacard it works out to be good value for money. Getting back into the city from Gogar by car during rush hour can be horrendous, I work near there and have got rid of my car. The trip out in the morning never seems quite as bad though.
Harrison Park has a playground, much smaller than Saughton but a playground nonetheless, with a canal along one side, canal towpath extends for miles and is a very popular walking/cycling route, recently scrubbed up with millennium money and plenty of ducks and swans for kids to see. I wouldn't aim to live in Gorgie/Dalry which are both local to Harrison Park, but Shandon, Craiglockhart, (closer to your target area) and Polwarth, Merchiston (further from your target area) are also close and I would be happy to live in any of them. Bits of Morningside and Bruntsfield close to Harrison park and very desirable but getting away from the city centre/Gogar line.
North of your city centre/Gogar route there are some quite posh expensive bits, I'm not so familiar with that area, but have been to Ravelston Park and it was pretty good. My brother lived over that way for a while, so there must be more affordable streets, like I say, don't know it so well.
This book is well worth it:
www.amazon.co.uk/Edinburgh-Under-Fives-Family-Friendly-Parents/dp/0955516137/ref=sr_1_1?ie=UTF8&qid=1459337872&sr=8-1&keywords=edinburgh+for+under+fives
Lists lots of stuff to do indoor and outdoor in Edinburgh, including quite a few but probably not all playgrounds. DS complained bitterly when I bought it as he is not under 5, but the content is still relevant.
The Meadows is a huge green space close to the city centre and has two play parks, one to the west of the Meadows which quite modest and one to the east of the Meadows which is another humdinger, get the bus and take a picnic job. Most of the housing roundabout there is tenement flats and its further from Gogar, but as a park it's worth a mention.
Edinburgh is fab, good luck with the move. If your kids are school age it's worth trying to find out which schools have spaces for them but you can refer to most of the preceding Edinburgh board for details on that! I check the Edinburgh board quite a bit and it's nice to see a change from the private school debates.