Systems to play as black?

Is there a solid system for black to play against openings other than 1. e4, d4, c4, or Nf3? I can only get a few moves into these less common openings before I find myself in really unfamiliar positions. I've been thinking about playing a pirc or KID setup against unorthodox openings, what are your recommendations?

There is no simple answer to your question.

Each of them needs to be taken into account individually.

For example, against 1.b4, Black's best move is 1...e5, and if 2.Bb2, taking on b4 (the "Exchange" variation, per se).

Against 1.b3, Black's best line is 1...e5 2.Bb2 Nc6 3.e3 d5 4.Bb5 Bd6 5.f4 Qh4+ 6.g3 Qe7 7.Nf3 f6 8.fxe5 fxe5 9.Bxc6+ bxc6 10.Nxe5 Nf6! 11.Nxc6 Qe4 12.O-O etc - I'll let you research the rest.  If you don't research it yourself, you won't force yourself to learn if I write out the entire line here.

Against 1.f4, 1...d5 is the best response with an immediate 2...g6 (avoiding lines with b3 and 2.b3 immediately isn't good - again, I'll let you do the research, but it has to do with White having to play Nf3 and e3 before b3 due to resulting pawn weaknesses on e3 if White plays b3 too quickly)

Against 1.Nc3, against, 1...d5 is considered Black's best.

Etc Etc.  So there is no "catch all" system that can be recommended!

Usually when responding to an uncommon opening it's best to either use what you play against 1. d4 or learn a specific line for the opening.

For example, I play the Slav against 1. d4 so I try for a setup that is like the Slav against 1. Nf3, 1. g3, 1. b3, and some other openings. However, there are a few openings where Black can often get more by playing another line, e.g., against 1. f4 openings Black usually should play g7-g6 at some point, a move which is not so common in the Slav.

Alburt's "Chess Openings for Black Explained" provides a complete repertoire for Black including several chapters on oddball first moves for White.

You can ignore the mainline repertoire and just use the oddball stuff.