Couple things to bear in mind choosing a rifle:
1. quality stock. rigid and stable
This means a quality synthetic or a laminate. No flexible junk. Squeeze the barrel and forend together. If the forend flexes, it's junk. I know you're on a budget. But starting with a junk stock won't save any money.
Both Savage and Remington use quality HS precision stocks, quality laminates, and cheap plastic junk. The cheap ones are black. Tikka uses a very rigid synthetic stock.
2. Good trigger. All three companies offer good adjustable triggers. The Tikka trigger is a good one. The new Remington xmark isn't bad. And the accutrigger is great. Shy away from older remington and savage triggers, there's only so much you can do with them.
3. Accuracy. Tikka offers a 1moa guarantee. The Savage and Remington will likely shoot this well. However, they aren't offering any guarantees. All three have great reputations. Tikka is the only one I haven't seen the occasional "what the hell" on. Mabey 'cause I've seen far fewer Tikkas.
4. Platform for a custom. Simply put, if you want to build a custom off the action at a later date gunsmiths prefer Remington. Savage you can rebarrel yourself.
5. Weight. The traditional mindset is lightweight is best. This isn't so for long range. Weight equals stability. However, you still have to pack it. The synthetic stocked Tikka is rather light. You decide a comfortable weight for you. Just bear in mind long range is a bit different.
So, The Tikka in synthetic or laminate. A savage in an HS synthetic or laminate, with the accutrigger. Or, a Rem in laminate or an HS synthetic, with the new trigger. I've had very good luck with CZ also.
For optics you need:
1. nice knobs! well marked whole numbers, ending in whole numbers, ellevation hash marks
2. side focus or paralax
3. enough vertical adjustment (40 inches minimum)
4. accurate tracking
Frankly, this is a tall order for budget priced scopes. I have tried Nikon Buckmaster 4.5-14, Nikon Monarch 4-12 & 4-16, and Bushnell elite in the lower price range. Of these, the Bushnell seemed the nicest. While I have never tried a Weaver, the accutrac system is reputed to be superb. I would check these out too. The leupold vx3's are a bit more expensive. They have better glass. In my experience, they don't track any better.
4-5 to 16 - 20 is perfect. 40-50mm is perfect. Vaguely remember Leatherwood. No details.
Caliber.
This is basically a trade off between muscle / range and recoil. This should play a role in rifle selection as you may want to pick a heavier rifle to help reduce recoil. I'll come right out and say I shoot pretty well. That said, the largest unbraked rifle I own is a 270win. As you are on a fairly tight budget, I'm assuming you want something you can shoot unbraked (at least for now).
I want to be able to shoot a rifle enough to become truly proficient with it. I also want to be able to spot my own shots. You can't do iether if you're getting pounded. Take my word for it.
As a mostly used for deer and coyotes with the occasional elk or black bear thrown in the 260rem would be a mild mannered, fun to shoot, and economical choice (those high bc high sd bullets make it a surprisingly effectice long range caliber). The next step would be a 6.5-284 (a real sweetheart). I seem to recall Savage is chambering a few? The ol' 270 would of course get 'er done. Bringing us to the threshold of comfortable shooting, the 7rem mag and 7 or 270 short mag.
Personally, if it's bigger than this, it needs a break or a lot of weight. Remember, this isn't typical hunting, the name of the game is accuracy. Comfort equals accuracy. I'll take a smaller lighter recoiling round I can shoot over a thumper every time. If you put the bullet through heart and lungs it will die, now!
For a mostly target / coyote / deer round I'd pick a 6.5. For an all around hunting round I'd go with a 7wsm. I wouldn't shoot any of the 30 mags without a brake. And you'd have to push a 30 cal 210 berger pretty hard (meaning recoil) to match the ballistics of a 7mm 168 berger with a lot less recoil.
