We got ours from the tourism office by the bears, but I just found the map as an image online here : http://spieleninbern.ch/images/dok/G...n-2015-web.gif
(Ours looks a bit different but overall the same - I really recommend getting a paper one).
The three favourites for us were:
1) Schutzenweg spielplatz
-- adventure/wild playground, old airplane cockpit to play in, old car, swings in shape of boat, lots of 'found' and created objects. Was not my favourite, especially some of the smells, but very different from any playground we've been in before and my kids loved it. Toilets on-site.
2) New outdoor playground at Westside shopping centre
. Bonus - fairly close to toilets inside the shopping center.
3) Elfenau playground
. Water play, big swings, climbing structure, smaller and bigger slides, and next to lovely Heirloom gardens. Toilets are about a 5 min walk and there is a sign to please not pee outside somewhere.
My kids are older tho, 8 and 5, so I wasn't looking for younger kids. Elfenau for sure has great play for younger kids. Gurten too (and it took me a while to figure out if I had a monthly, weekly, or daily transport pass then the trip to the top of the Gurten is included). We don't live in Bern unfortunately but have spent many, many happy weeks there.