So as far as I understand everything in our solar system orbits the center of mass (Barycenter?)

 

So in a binary system with planets how would the planets orbit?

Would it orbit around both suns in one long orbit?

Orbit around one sun?

Or orbit kind it like a figure 8?

 

I'm no science expert just an enthusiast.

it is most likely that the planet would either orbit one star closely or orbit both stars like they were one big one (put simple).

a figure 8 pattern would be slingshot out into space to become a rogue planet.

even if the math just so happened to work out it would only last a short time as solar wind, star fuel consumption, and many other variables would quickly imbalance the integrity of such a spectacle.

Romron, check out this link, written for non-astronomers by a NASA astrophysicist. It describes stable binary star systems with planets.

 

http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/980122c.html

A possibility is that the two stars are orbiting very close to each other and the planet is much farther away from both, so its orbit could be pretty much stable