Can I transfer achievements/games from my old gamer tag to my new gamer tag?

Sorry, but no. Nothing can be transferred between accounts. You would need to use your 360 account on the One if you want to keep the achievements, etc.

If you want the gamer tag you can change the Microsoft accounts around. However you will lose all achievements etc from the other account. You'll need to make a new email with Microsoft transfer your old profile to that one then transfer your old email to
the new profile with the desired GamerTag. Otherwise you'll need to change the gamertag on the account with the gamer tag you want then change it to the gamer tag you want on the account you primarily use. Changing your GamerTag costs $10 per switch. So all
and all it'll cost you $20. Good luck my friend.

It greeted me with my new gamer tag when asking me if I'd like to transfer my Xbox 360 settings to the new Xbox but still shows my old gamer tag in the top left corner with my email And gamer points ect

Gamer score leaderboard also shows as my old gamer tag above my game points, are these places where my new gamer tag should be on Xbox 1? On my Xbox 360 my old gamer tag is nowhere to be found even if I look for it

For some reason my new gamer tag linked itself to my old gamer tag email address. I don't know how to change that and to link my old gamer tag back to my Microsoft account. I believe that's why I cant play as my old game tag. Is there a way to fix that?