How do I add two secp256k1 keys together? I have two public/private keypairs, A and B. I want to add them together to get a new keypair AB.

I'd also like to be able to add the public keys of A and B to get the public key of AB.

How do I do this?
Most ecc libraries will have this function, but if you want to program it yourself, here's what you do:

First, compute the slope of the line containing the points A and B. Let A = (X_a, Y_a) and B = (X_b, Y_b). The equation for the slope is:

s = (Y_a - Y_b) / (X_a - X_b)

The resulting point, we'll call C = (X_c, Y_c) = A+B. Doing some math, you get:

X_c = s^2 - X_a - X_b
Y_c = Y_a + s (X_c - X_a) = Y_b + s(X_c - X_b)

If X_a == X_b, then it depends on Y_a and Y_b. If Y_a == Y_b, then A and B are the same, so really, you're just computing a point doubling (2*A). If Y_a == -Y_b (the only other possibility) then A+B = the point at infinity, or the identity. Usually, that's not a very interesting point for cryptography. Computing 2*A is a little trickier, but can be done. You're already doing that when you compute the public key from the private key with G anyway, so I'll assume you have access to something that lets you double a point.

Note all the operations are all field operations, so you have to mod by P for secp256k1 (FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F in hex).