A router will be your BEST friend!
I haven't inletted any airgun stocks yet, but I've done plenty for 10/22's and a few others. For stuff like the trigger opening and the stepped parts inside the inletting I used to use a drill press & a forstner bit just slightly smaller than what the finished cavity was supposed to be. Just drill, move, drill, move etc... Just nibbling along then use a nice SHARP chisel to clean up the corners.
For the longer parts I used to measure the depth of the barrel channel (or chamber channel in this case) and set my table saw to a hair under that depth, set my fence & rip down to my stop point. You can route a barrel channel with a table saw, but it'd take a good amount of time. Hence the 1 inch cove bit & router! hehe
Good Luck!!! and most importantly HAVE FUN!!!