So I am just making sure I am on the right track with this.
I have the recurrence:
T(n) = 2T(n-2) + 1
I am trying to solve this recurrence to get the time complexity
T(n) = 2(2T(n-4) + 1) + 1
T(n) = 4(2T(n-6) + 1) + 3
…
T(k) = 2^k T(n-2*k) 2^k -1
Am I on the correct path so far? Where do I go from here how do I find the time complexity. Please explain it and don't just give me an answer I really want to understand / think in the right way to get to the solution.
First of all, in order to solve this explicitly you will need to know what $T(0)$ and $T(1)$ are.
There are two reasonable approaches: first, you can just write down $T(k)$ for, say, $k$ from $1$ to $10$, see what the pattern is, and move on from there.
Second, you can use generating functions: define $f = \sum_{i=0}^\infty T(i)x^i$; then your recurrence gives $$f = T(0) + T(1)x + \sum_{i=2}^\infty (2T(i-2)-1)x^i.$$ You can rewrite this in such a way that you get $f$ on the right-hand side as well (I'm being vague here because you said you wanted to work it out for yourself); solving for $f$ will give you an explicit formula for $f$ as a function of $x$. Then $T(n)$ is just the coefficient of $x^n$ in the corresponding power series.