Big Z

With the added info of: Handloading; No magnums right now, but getting one later for elk anyways-- I'd give a cartridge range of 243win - 30-06sprg. If you MAY need the rifle for elk, I'd bump the low end up to 260rem. My personal choices from this range: 243, 260, 7mm-08, 280.

The coyote skins are a challenge. The 243 is the best choice in the given range if you take your yote skins seriously. At the same time it's a proven deer round and one of my personal favorites to target shoot to 1000yds & beyond (not beyond by much though). It'll give the lowest barrel life of the short action choices, and many, if not most factory barrels won't shoot the big 105+gr VLD bullets. This decreases your choices of long range shooting bullets, in turn making the 243 more difficult to use at a distance. If you care less about coyote pelt damage and more about long range shooting with the possibility of elk, I would eliminate the 243.

In the short action realm, the 260 and 7mm-08 are better suited to elk (note caliber restrictions in states you may hunt elk in; a 243 can be & has been used in some states) and long range shooting, but will suffer from a lack of loading options for taking yotes less much pelt damage; if there are good yote options for bigger rifles like these, I don't know of them. I just stick to smaller rounds and haven't cared to do any R&D.

Now the 280. Add velocity to the 7mm-08 and voila. This will be the flattest shooting (bullet dependent) option for commercial cartridges in the given range, and suffer the least from wind drift. Useful for long range shooting. On the negative side, you will add weight from a long action, and 2-4" longer barrel. Something to consider if you do much walking when hunting. The 7mms are my favorites for long range shooting, and I've a healthy supply of 162gr amaxes for slingin' em far. These bullets haven't been produced in awhile but will be coming back according to Hornady. The 26cal may or may not be the better option for affordable and available long range bullets at this time.

I'm not much of a fan of the 30 calibers because they don't offer much for higher BC bullets for long range shooting until you get very heavy, and that makes them the most expensive for a given manufacturer. However, the 308 & 30-06 are obviously proven do-all rounds. The 308 is most noteworthy for its significant barrel life; I've read many reports of several thousand rounds. The money/time savings on barrel changes are something to consider.

Rifles. I prefer Remington 700s and Savage 11/111s. Savages have good aftermarket parts availability and offer easy in-home parts changes, including barrels--which are much more expensive and time-consuming on the Remmy with a 'smith involved. I'd choose a Savage if you like to tinker. Remingtons have the best aftermarket parts availability and IMO are more rugged. There are also barrel nut options out there for the 700s these days, and I've read promising results from some Remage nut users. I'm not sure I'll ever do it myself, but the option is there. On the Savage note, you can get a Stevens 200 cheaper. Essentially a plain Savage, they use the same parts. I've done a couple builds on 200 actions and they worked well. Several other makes and models of rifles out there are good options. I've just gone over what I prefer. Although heavy rifles are more ergonomic and solid for long range shooting and handle the heat of higher volume shooting better, I'd stick with buying a sporter weight rifle for now. This keeps your budget better balanced for optics & reloading supplies. If you want to, you can make changes towards a heavier gun with higher quality parts than most factory configurations later on. The guns usefulness for hunting goes down as weight goes up IMO, but I'm in mule deer country and have to walk a fair amount at home and am only a 2hr drive from elk hunting, which is about 2-3x as physically demanding.

Optics. The addition of long range shooting demands a scope with repeatable adjustment dials. I run Leupolds, and have 7 with turrets. My favorite do-all is a 3.5-10x40 with M1 elevation and windage dials. Buying used, you can expect to spend 400-550 on a vx3 or vari-x iii after sending them in for turrets, depending on the age and condition of the scope. I have 2 M8 6x42s with M1s. Also good options IMO, and save weight. They can be had for 175-250 used, and you'll have about $150 into putting M1 ele/windage dials on. I know there are other good options, but these are just what I have experience with and what should also fit in your budget.
A rangefinder is one of my most valuable tools. For the budget, I'd look into Bushnell Elites (1500, 1600arc, 1-mile arc). Used will help the price. They're all good enough to read hunting ranges in the field but higher end models are better for picking up readings on long range targets.