Coyotes, deer, target shooting, in a bush gun package...
That's certainly a tall order.
Well, for coyotes, one should prefer a lighter, higher velocity round, a varmint round. It should come in a package that's accurate at long range. Similar concerns are present for target shooting. The problem with this is that varmint rifles tend to be long, and light rounds are more easily thrown off course than heavier rounds (due to inertia).
To me, a bush gun is a short, manueverable gun that's pointable and fires a heavier, slower rounds. So, really, a firearm that can be used to hunt both coyotes/target shoot as well as be used as a bush gun is, as I said, a tall order: the concerns are nearly opposite. Unless, of course, coyote hunting to you is not done on large, open plains, and you don't plan to do any real, long-range shooting.
I think you do have a few options, though.
If the bush you're going to be experiencing isn't all that thick, a high velocity, light round may not be a bad choice. It really depends on how thick of bush you're going to be taking on is. A long-barreled rifle chambered in .25-06, .243, or .270 may be a fine choice, again, if you're not going to be taking it through really thick bush.
If you're going to go through really, really thick unbearable stuff, as others have said, a .30-30 leveraction rifle is a superb choice. But you will not be able to do any real, long range target shooting with it. A CZ 527 or a .308 or .30-06 rifle in a bolt or lever-action package with a short barrel can be a fine choice, and these offer you more versatility when it comes to long range shooting. Of course, you also have the option of going with a 12 or 20 gauge loaded with slugs, or a handgun chambered in a nice, powerful caliber--10mm, .44 Mag, .45 (Long) Colt, .454 Casull, .50AE, .460 S&W, or even the .500 S&W beast. Obviously, target or coyote shooting options with these are extremely limited or non-existent. Still, handguns are small, offer great pointability, and light weight. A scoped Thompson Contender, a revolver, or heck, a Desert Eagle could be a fine choice if you want to do close range bush hunting. And then, there's the possibility of going with an AR-15 platform rifle. Chambered in .50 Beowulf, .458 SOCOM, .450 Bushmaster or some other close-range, hard-hitting caliber, I'm sure you'd be proud. Flat-top, flip-up sights, decent glass, a nice stock... yep, I'm sure you'd be happy, and, at will, you could easily go to a longer, heavier freefloated upper in .223/5.56mm for nailing coyotes or targets at longer ranges.
If you really must get a do all gun, I suppose your best option is a .308 or 7.62x39mm carbine in a bolt or leveraction package. Perhaps a Saiga or PTR or M1A/M14 rifle could be good, their operations are proven and robust. That, or a good AR would suit you fine. But your concerns are very nearly opposite, like I said, so if you only get one gun (or, in the case of the AR, one upper) you will be trading strength in one field for weakness in another.