the 22 issue . i guess its up to you . do you need to re-familareize yourself to shooting ?
if so a 22 is a great way to do it .
good calibers to choose from are 243 , 270 , 308 and 30-06. these are very common calibers and will serve you well . a good reason to choose one of the above calibers is ammo availabilty . you can find a good selection of loads at a good price almost anywhere they sell ammo . the 308 can be purchased extra cheap in mil-surplus .ammo can be expensive but most guns can be shot for less then a doller a trigger pull .to compare prices go to "ammoengine.com"
what action gun will you choose ? semi auto loading , lever , pump ,bolt ,single shot are most of your choices and will help decide what brands to look for . you stated your on a budget so maybe a used gun is a good choice . remington 770 and savage 110 are also being offered new from the factory in package guns at good prices .
for tons of good reading on hunting and shooting go to "chuckhawks.com" then click guns and shooting.
hope this helps ,good luck .