Well that's why we're talking! You seem like a very responsible young man.
If I remember correctly NY is shotgun only for deer. That said I still like (and own) a NEF Handi Rifle and my brother has two, one for himself and one for his son (Pa hunters). My hunting partner and sons (2) have one too. We all have multiple barrels for each including shot gun barrels (Cal hunters). Mine started as a 25-06 (30-06 necked down to 25 caliber) for varmint Your nice Rem will be a one trick pony, it can only hunt one thing in NY, varmint. Get the .243. You don't walk around all day looking for yotes or ground hog so a varmint barrel won't hurt but I'd still stay with a sporter just my opinion.
Good optics are never cheep but are worth every penny. A Leupold VXII should be close to $300. There are some good scopes for about $200 but I haven't tried them.There are also some good used scopes out there like Redfield and Weaver but research before you buy. Bushnell makes some great scopes but they also make some junk. You get what you pay for.
Back to what I said before, the NEF has many inexpensive barrels including rifled slug, muzzle loader and turkey barrels. They are easy to change with a #3 screw driver. With quick detach mounts a good scope can be switched to a different barrel or you could use open sights. A quick trip to the range to sight in and you're ready to go on any hunt that's available.
Go to the NEF forum and ask some questions. Look at the NEF classified forum. If you buy a NEF Handi Rifle, have the trigger done right and a stock that fits with a good recoil pad, you'll still have money for a scope and slug barrel. If you got their muzzle loading barrel it extends your range and season with less people in the woods. The major plus is one frame can be fitted with a rifle, shotgun, slug, or muzzle loading barrel. No barrel will cost more than $100. No they're not pretty but I and four people I hunt with one one
IMHO don't limit yourself. If there's hunting to do make sure you're available to go with a gun of your own. There's a very old saying, "Beware the man who owns one gun, he probably knows how to use it."
Three other things 1) There are other forums on this site visit them and read them. Nothing excites a man more than passing what he knows to the next generation and this is a great web site for that.
2) A quote from Clint Eastwood, "A mans got to know his limitations. " There are allot of folks here that love and have the money for fine guns. There are others that don't. You may have the means to get a slug gun, varmint gun,shotgun and muzzle loader. I don't know, but think it over and "Know your limitations". I had to work all summer to earn the money for my first deer rifle, a Winchester that I still own. If you have money to burn buy an Encore. When you have one shot and one gun you know your limitations.(There's a T/C, Encore forum here too.)
3) Talk to the people that will take you to hunting, they should know their limitations and yours too. They'll be happy to add their thoughts and hunting experience. None will laugh when you show up with the same gun for rabbit, deer, varmint and turkey because it will be the right gun for the job.