There is nothing wrong with the Weatherby Vanguard. It's a very good rifle for the price. The only possible negative is their weight, which is a bit on the heavy side. But of course that can also be a good thing, as it reduces recoil. I have one in 30-06 which I bought new a few years ago for $320. I did install a new Timney trigger though, as the stock trigger was way too heavy. Cost was about $120. So I guess that makes two negatives for the Vanguard. A Savage would be a good choice. They have what they call the "accu trigger." It is adjustable so that you won't need to pay a gunsmith to adjust or replace your trigger. Here is an article on the accutrigger: http://www.shootingtimes.com/longgun_reviews/savage_0813/
As far as hunting for squirrels or rabbits, honestly there is very little opportunity for either in this state. The eastern gray squirrels live in cities and parks. Generally there aren't anywhere you can hunt. I'm not sure about fox squirrels...I didn't realize they were legal to hunt. But most of the squirrels you'll see out in the woods are protected species. Ground squirrels can be hunted but they would be mostly in eastern Wa. You will see rabbits occasionally out in the woods. The only ones I ever see are on the edges of logging roads. I would suggest spending your time mostly scouting for deer and elk for next fall, and carry your 22 rifle if you wish for any opportunity that may present itself. Grouse opens September 1st and you can hunt them with your 22 rifle.