It's your first rifle and you say your going to be hunting deer and coyote's. Not a big fan of the 243 on deer but I have shot a few and it does work better than I'd admit. But hunting what your going to be hunting, you should have more shooting at coyote's than at deer. With that in mind, I'd say get the 243. If you decide later to hunt something larger than deer then think about a newer rifle in a more apropreate cartridge.
I have a 25-06 that I've been shooting for about 20yrs now. It does a great job. But I'd stop at suggesting it would work well on elk, to many "if's". If at some time you do want to hunt elk, I'd suggest the min caliber would by 6.5 and a 140gr bullet if you aren't reloading by then. The 270 and the 7-08 are certainly adequate cartridges but light as they are on recoil, they would be far to violent to me for a coyote cartridge.