For me, I’d go with a Ruger American Rifle (just the plain Jane one, not the predator or any of the more expensive versions) as an introductory budget hunting rifle. I prefer 308 as it is a short action and I prefer shorter rifles. Both 308 and 270 are excellent hunting cartridges. 270 has more horsepower (via larger case capacity), but also more recoil. 308 shooting 150 grain bullets is fairly mild recoiling, but in a light rifle will still let you know when it goes off! As a beginner you might want to look into 243, 6.5 creedmoor or 7-08 as they are all lighter recoiling. If you know anyone or can find someone that will let you, I would recommend shooting one or or both (308 and 270) before purchasing one, just to see if you are comfortable with the recoil.
Ruger American pros:
-tang safety (easy to operate and get familiar with).
-70 degree bolt lift (you can mount your scope lower to get better eye to scope alignment).
-Ruger has updated the magazine to a double stack design that is supposed to be more reliable than the earlier rotary design.
-Ruger has a good reputation and a lifetime warranty. I’m not aware of any other rifle manufacturers that have that.
Cons:
-the safety does not lock the bolt closed when engaged. Not a huge issue, just something I prefer.
-price, it is one of the pricier budget rifles.
There are also the Savage Axis’, the Remington 783, Mossbergs and Thompson Center Compass (I think that’s their cheaper one?). I have experience with the Savage Axis line. They are cheap and accurate but for me have the ergonomics of a boat paddle that slings projectiles. My cousin has a Remington 783 in 30-06 that he takes a lot of game with. He likes it, I dislike where the safety is (preference thing). I’ve handled the mossbergs, the actions don’t feel very smooth and seem prone to binding if run hard and not perfectly straight. No experience with the Thompson Centers, but have read that they are accurate.
That’s it for my experiences and opinions. Now, what you should do and what matters:
Go to some gun shops and handle as many rifles in your budget range as you can. Work the actions. Hopefully the gun shop will let you dry fire to get a feel for the triggers. Work the safeties, see what you like. Get some scopes mounted, check for scope to eye alignment (comfortably shoulder the rifle with your eyes closed, then open your shooting eye, the crosshairs should be perfectly aligned with your eye). If you have to wiggle your head around to see the crosshairs clearly, the cheek weld is not correct. To correct that you can get higher / lower scope rings (bolt / bolt handle clearance allowing) and or cheek risers. Make sure the length of pull (distance from the recoil pad to the trigger) feels comfortable (you don’t want to feel like the gun is cramping your style or that you’re reaching way out for it).
Glass. I like Vortex, many do not. I like scopes with drop compensating reticles. I have two vortex diamondback 3x9-40 scopes and they take game just fine. They do have a shortish eye relief (distance from your eye to the ocular lens), which can be an issue with recoil (scope eye). I have never had that issue with those scopes, even on my 308. I would not buy a vortex below the diamondback line. Leupold, Redfield (made by Leupold) and Burris are also good scope manufacturers. I would recommend buying a scope within your budget that has lifetime warranty. I haven’t needed it with my vortex scopes (even the cheap ones, and they get knocked around!) but it’s nice to have.
There, that’s my novel, I hope it helps. I am a true gun nut, specifically hunting rifles. A lot of the stuff I mentioned is stuff I learned along the way that I wish someone had told me before hand.
Happy hunting!