Where to publish new mathematical identities?

As Peter Heinig commented, if the mathematics behind the identity is novel and important enough, then you should select a journal like you would select a journal for any other paper—if it's a combinatorial identity, look for a combinatorics journal; if it's a number-theoretic identity, look for a number theory journal, etc.

For the specific case of constants such as $\pi$ and $e$, most novel identities for them are nowadays discovered with significant computer assistance. The journal Experimental Mathematics is one place where such identities have been published, e.g., I'm fond of Jesús Guillera's paper About a New Type of Ramanujan-Type Series, which contains some amazing identities such as the following one due to Gourevitch (which I believe is still open as of this writing):

$$\sum_{n=0}^\infty \frac{1+14n+76n^2+168n^3}{2^{20n}}\binom{2n}{n}^7 = \frac{32}{\pi^3}.$$

For new combinatorial identities in the spirit of S. Ramanujan's work, there is the Ramanujan journal.

The American Mathematical Monthly, Mathematics Magazine, and The College Mathematics Journal (all published by the Mathematical Association of America) might be appropriate.