Welcome to the forums! I'm not a harmonica player but I have taken a swing at it and I remember a little bit about "straight" and "cross" harp. As you suggested, it just comes down to knowing the key of the song and making your choice based on that.
A straight harp would be using a harmonica in the same key of the song; so if the song is in C major you would use a C harmonica.
I believe with cross harp you're playing on a harmonica that's a perfect fourth higher than the key of the song; so if the song is in C major you would play an F harmonica.
As you suggested, you can use the circle of fifths to identify which harmonica to use in a "cross" harp scenario. Take a look at the circle of fifths:
If you read it clockwise it ascends in perfect fifths (hence the "circle of fifths"). However if you read it counter clockwise you get perfect fourths, and that's what you will use to identify which harmonica to use for cross harp. You can see that F is immediately clockwise of C, so F is a perfect fourth above C.
Regarding the inner/outer circle; the outer circle is all major keys and the inner circle are all the relative minor keys, meaning the minor keys which share the same key signature as their respective major ones.