Honestly, you should scroll through the topics in this forum and its' 2 reference sub-forums. Each forum will have some of the information you're looking for and in some of the reference forums, you will find a great deal of technical info that is very useful in making these kinds of decisions.

Your profile doesn't list your state and that can have a large impact on what is legal in your state and thus what members might recommend for you. You also didn't mention any of the terrain, distances, etc. you face or what type of hunting method you're considering (i.e. stand hunting, spot and stalk, still hunting, using mouth, hand or electronic calls, etc.). If you could post this information, it would help members give you more precise suggestions and recommendations.

In general, you could use almost any caliber for a ground squirrel or woodpecker but would want something with more power for a coyote. It's deer and bear season right now so most people are using calibers they use for that game to shoot any coyotes they see. If I were only hunting coyote, I would use a .223 as it offers a lot of performance over a pretty wide range (not all coyotes come in close).

Hopefully this helps some and welcome aboard.

p.s. I forgot to mention that there really isn't one single best varmint rounds for all varmints and sometimes not even for the same varmint. depends on the terrain, conditions, type of hunting and the hunter.