Honestly the list of not to buys is shorter than should buy. Everyone has their favorites. If cost is the primary driving factor, find a dollar amount they can afford, ($550 CAN let's say). Then go with a common caliber with reasonably priced ammo that will take the game you want (30-06, 270 medium to larger game or 223, 22-250 for smaller game).
Last piece scope (or no scope in the case of a light lever 30-30 say). There are other things to consider but but let's stop there. A good quality scope is important. Here again afford what you can, you can always upgrade later.
Me/myself/personally in the "budget" rifle category, I do have several Savage Axis II. These rifles for me represent calibers that I want to own and shoot but I may not take hunting. For me (and I do reload), each and everyone of my rifles shoot .5 MOA or better. There are a ton of aftermarket replacement barrels for Savage (side note nothing against the RAR but if you're interested in barrel swaps these tend to he pricer but still great rifles). There are aftermarket stocks and chassis systems. The Axis II cones with an AccuTrigger that is adjustable and has a clean break and if you don't like it you can always upgrade.
I would be confident in killing an animals with any of my budget rifles. YMMV
HaYen
