I know that it should be either a sphere, torus, Klein bottle, real projective plane, or a connect sum of any combination of these, but I don't know the steps in identifying what kind of surface this is.
I know there are $2$ boundary components, but I don't know how many vertices or edges there are, or how many Seifert discs. There's an even number of half-twists, so this surface is orientable.
With a couple of moves, if you know how to recognize it you can see that it is a twice-punctured $T^2\mathop{\#}\mathbb{R}P^2$. If you do not know how to recognize it, I have draw some pictures that might convince you.