I know I need to use $1+\tan^2(x) = \sec^2(x)$, but I don't know what to do next ? How do I get my $U$ variable so I can use $U$ substitution?
I would let $U:=\tan(x)$, say. Then, as the comment says, $dU=\sec^2(x)dx$.
So, we have $\int\tan^6(x)\sec^4(x)dx = \int U^6(1+U^2)dU$.
Since the derivative of $\tan x$ is $\sec^2x$, you want to have all tangents except for one factor of $\sec^2x$ that will be part of your $du$. Use the identity that you mentioned to convert two of the secants to tangents:
Now let $u=\tan x$, $du=\sec^2x~dx$, and your integral becomes