I grew up in Western Oregon. Now live in Central--Redmond.
There are many places to hunt over there. Elk, deer, cougar, bear, coyotes, rabbits, nutrea, .....squirrels, birds.
I used to work for the ODFW there at E. E. Wilson north of Corvallis, raising pheasants and "improving" the place for the public. When I was younger, not too many people knew about the rabbits there. Best rabbit hunting anywhere!--and it was open to roam and shoot! That place has changed a bit much since my teen years.
You, living in Eugene, are in the middle of excellent areas to hunt critters and birds.
The McKenzie unit to the east is full of big game. All of the coast range to the west is teeming with wildlife.--and much of it is open to hunters.
I was up the road from you in Corvallis for a few decades--spent most of my free time hunting the valley and the woods. Things have changed a bit since I was there, but it is still a wonderful area for hunting.
We have plenty of animals over here too.
I have three sons who have recently left the nest, so I know how important it is for you and your son to get out together.
PM me and I can better help.
I'm new at posting on this site, so if you PM and don't get an answer, let me know on this thread. I'll help.
Also, do you know of this site?
http://ifish.net/board/index.php
Sign up, go to the Hunting area, and ask there. I'm certain you will find good people in Eugene to advise you, maybe even take you out!