Re-enlistment bonuses are taxed like regular income and are typically added in with your regular wages and reported on the same W-2.   You will also have taxes automatically withheld from your bonus, generally at the 25% or 28% rate. This is automatically done by the government, and not something you can change. If the withholding is too high for your tax bracket, then you will likely receive a larger than normal refund the following year.

Did you happen to be in a combat or tax-free zone when you re-enlisted?   This would make your re-enlistment bonus tax-free.  

You can check with your MyPay account to verify a second W-2 was issued.

If you're unsure if you're W-2 is reporting your re-enlistment bonus correctly, before filing, go speak with your finance office to verify.