OK, you are starting as a new hunter. You are in the kind of environment where 100 yds will be a long shot due to brush and scrub, etc. 30-30 is fine and so is 35 Remington. Marlin 336 is fine and so are all it's store-brand siblings like the Glenfield.
Scope is a good idea, but you don't need much. 4X fixed power with "duplex" reticle will get you on target quick, have plenty of field of view, have way fewer moving parts to go haywire, etc. They are dirt cheap used. You can get a decent scope made in Japan back in the day (say 5~10 years ago) for $25 off the 'Bay. If you get a nice rifle with decent blueing, you can look for an older Weaver steel tube K4 and they will match up pretty well. The older Steel tubes are hell for tough and cheap enough too. You can always upgrade your scope when you want.
For a first hunting rifle I'd be looking at a used one from a gun shop or a pawn shop. There are tons out there right now. Find one with nice fit and finish, maybe with rings on it or even a decent scope and spend the rest of your modest budget on ammo and range time. Get used to shooting from standing, sitting and kneeling. Those are the positions you will use afield. Check over the scope base(s) and make sure they are down well. Might want to rebed them with a dab of JB Weld under and some Loc-Tite on the screw threads into the top of the receiver - loose scopes do not work well
If you run across one, do not turn down a Browning BLR is say .243 up to 30-30. They are fine rifles, hold their value well and shoot well. Same goes for a Savage 99. If these sorts of rifles don't show up while you are searching, they may in "hunting camp". They are always a good investment.
For a first timer, until you get used to recoil and such, I'd shy away from 30-06 or 270. Get some center fire range time in with your rifle, and maybe a few of your buddies rifles. Once you are comfortable with them, you can add a second higher powered rifle to your collection when a good deal comes along
Nothing wrong with also having a 22 to use as a skill builder. You can get a Marlin 39 lever in 22 so the ergonomics are similar and put in a lot of time at the 50 yd range from the field positions. The ammo cost savings will pay for that gun eventually