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I own a 700 VSF in .308 and love it. I've had it for about three years now. It's a very accurate rifle, and the .308 cartridge is alleged to have "inherent" accuracy. I don't know if that is true or not, but it does perform very well in my particular rifle. Not sure I would want to use it on dangerous game like an Alaskan brown bear, for instance, but I would be pretty confident of being able to take anything else with it as long as I do my part and make correct shot placement. Brand new 700s are almost certainly going to be well out of your price range, if that price you named on the 770 is your upper limit; but, you can probably find a used one for closer to that price point. They are durable rifles, and so as long as it has received decent care, a used 700 will perform just as well for you as a new one, and you'll have the satisfaction of owning a "finer" rifle than the 770.

Of the other two cartridges, I would almost certainly go with the .30-06. It will kill almost anything in North America. It has a wide variety of available commercial loadings. It is available anywhere that sells ammunition - year in and year out. It has 103 years of re-loading data available for it. AND (bonus points), it gives you an excuse to pick up that CMP Garand you've always wanted too.