Hi there, I spent 5 days in Aviemore last year, though had my own car to get about so am not sure about public transport, but I liked Aviemore, particulary around Loch Morlich and Glenmore and would think it an excellent base for at least 3 days.
I have recently climbed Bynack More, which I'd highly recommend. Check out the Report by Bod for doing it in the summer with the Barns.
There is a corbett, Meall a Bhuachaille there too and if you aren't going along the track for hills, the Green Loch is worth a wee stroll.
Obviously Cairngorm and Braeriach can be climbed from this area as well.
Check out this link for other activities and check out Loch an Eilein and if you go there look out for the squirrel in the car park. (not sure how easy to get there - perhaps hiring a bike)
http://www.rothiemurchus.net/index.htmlBraemar is a good area for hill walking, but many of the hills are accessed by Linn of Dee or other places that I am not sure are accessed by public transport. Lots of big days to be had in there. Wildcamping is a sure possibility and there is a place to hire bikes in the vilage of Braemar..... I know of two Hostels there - one is Braemar Lodge (I think) and the other I can't remember the name of....
I am sure there are smaller hills perhaps nearer to Braemar, but Aviemore would be my recommendation. Have fun!