Do NOT buy a Remington 770.....

My go-to coyote gun is a 243. I own a Remington 700 243, a Tikka Lite 243, and a Savage special run 243. They all shoot equally as good, buy whatever fits you the best. As for the scope a good scope that won't break the bank is a Nikon Buckmaster 3-9x40.

Here are a few rifles I would take a look at for reasonable prices, in no particular order.

Remington 700 SPS Stainless (the regular Remington 700 SPSs are prone to rusting)
Tikka T3 Lite - great gun for the money with a user friendly adjustable trigger, I really like the detachable mag options here.
CZ 550 - Great guns with a great trigger system
Savage - Lots of different options here, the synthetic stocks are flimsy, but they still shoot great, and the accu-trigger is a great trigger.
Stevens 200 - around 280-300 dollars, leaves money for a good trigger job (35 dollars) and higher quality optics
Ruger M77 MKII Hawkeye - kind of a hit an miss deal on quality trigger/accuracy. I've shot a few that shoot very well, but the triggers leave alot to be desired in my book
Browning A-Bolt - One of my all time favorites, will cost a lil more money than the rest of the above rifles.

Hope this gives you a good starting point, if you have any more questions feel free to holler at me.