For a lightweight walking rifle, I'd go for the Remington Model 7, a Ruger Compact, or the new Ruger American Compact. Caliber is up to you, but there are plenty of options that'll work just fine, with 308 or 7-08 being easy-to-get choices. Other rifle options include the Tikka T3 lite.
I've got the Ruger Compact in 260 and it's so light and handy that I hunt with it a lot. I think the rifle, without scope, is less than 6 pounds.