is my elk tag only good for archery or can I move into rifle season with the same tags
It depends on which Zone you purchased the tag. Check the rule booklet for your zone and follow the A-tag options across the page. Some allow early archery hunting and late muzzleloader hunting (Palouse Zone). Others like the Panhandle Zone allow early archery hunting, followed by "any" weapon hunting, followed by muzzleloader hunting, followed by more late archery hunting (what a deal!).