The Alvie and Dalraddy estates 4 miles from Aviemore are traditional Highland sporting estates, and they offer a range of field sport including deer stalking and grouse shooting. They provide other activities too such as clay pigeon shooting, archery, paintballing and quad biking.
The Rothiemurcus Centre claims to have one of the best clay-pigeon shooting facilities in the UK. Details can be found here:
And for more on clay pigeon shooting and archery: VisitCairngorms Field Sports
If you're up for a day trip, you could also try Bowhunter Archery. It's set in natural woodland, providing (a safe but fun) experience of Bowhunter Combat Archery, Alternative Field Archery & a 3D Target range. Its on the B9092 Nairn to Ardesier road (approx 15/20min drive from Inverness and a 5 minute drive from Nairn).