Go to Profile & system > My profile > Customize profile and select your current gamertag to enter a new one and check its status.
To find another person, Microsoft requires you to use specific social features rather than a broad search: xbox live gamertag finder
Forget unreliable “gamertag finder” tools. The best, safest, and fastest methods are: Go to Profile & system > My profile
Would you like a printable checklist or a short script to ask a friend for their gamertag politely? xbox live gamertag finder