Remington SPS is a great choice. I've used that platform for several builds. For a long range rig, go with the 243 or 308. Personally, I lean towards the 308 as it can take deer at longer ranges than the 243. Now I know you said 'coyote gun', but once you start shooting long range, the deer start to look good! But definitely shy-away from the high-power 22's. They might have some screaming velocity, but they don't have a good enough BC to go the looooong range.
The Rem700 is a great platform because every gunsmith knows how to work on it and everyone makes aftermarket parts for it.
In terms of optics, both the Bushnell 3200 and Buckmasters are good scopes at a value price. I've used the Bushnell 3200 10x and I have a cousin that shoots the Buckmaster 6-18x40 and the 4-14x40.
My recommendation...start life with the stock rifle and add the scope of your choice. Then when your budget permits, start upgrading. Here are some ideas.....
My nephew's rifle is an SPS in 270. Here's what was done with his rifle and it's what I recommend anyone do with a stock rifle....
Nephew's Transformed Rifle!
Next, here's my 308 that took a similar path....
My 308 1k Rifle - The Evolution Continues!
My 308 eventually turned into an all-out custom....
My 260 by MCR
Good luck, have fun and keep us posted!
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