The 7X57 or 6.5X55 (I'll assume this is what was meant, as the 6.5X57 is far less common) are not "more handloadable" than the .308 or .270. They simply require handloading to achieve the full potential of the chambering, due to neutered factory loads that are safe in old rifles of questionable strength and condition. Handloads in these chamberings allow the full potential of the chambering to be seen, provided a rifle of sufficient strength to contain the pressures they generate when loaded hot.
The .308 and .270 do not suffer this same fate, thus the gains in handloading are smaller, at least in terms of absolute velocity attainable.
If you spend your money wisely, a serviceable, if not fancy .22, a solid centerfire rifle and optics, plus a serviceable shotgun should be attainable on your budget.They may not be the prettiest or flashiest, but they will work just fine.
As to the recoil question, it sounds like you would be best to limit yourself to chamberings that generate moderate recoil, in order to avoid causing yourself discomfort. If you better defined your velocity expectations, you would get better answers. For the most part, high velocity chamberings capable of handling moose hunting tend to have moderate to heavy recoil. If you require light recoil, then your expectations of high velocity may not be possible without dropping to a very small bore diameter and bullet weight. Moderate recoil and moderate velocity (ie 7mm-08 is likely a much better choice, especially for a new hunter).
If the .338 Federal "kicks like a mule" to you, then I would say that at this time, your level of recoil tolerance is quite low. You will need to fire many rounds of lighter recoiling chamberings/loads and gradually increase to a higher level. For reference, a .338 Federal is approximately the same level of recoil as a .30-06, depending upon the exact loads chosen.
