What are examples of appropriate journals to publish new mathematical conjectures?
There are several ways in which conjectures propagate, and each has its own method for publishing.


As future work/further direction on a set of theorems you proved. In this case, you submit the paper based on the set of theorems, and the conjecture is just a bonus. Your primary contribution is the original theorems.
As a reasonable technical lemma to use to prove a difficult theorem. One again, submit as you would a paper proving any other theorem. Your primary contributions is the interesting/non-obvious consequence.
As a consequence or expected result based on a novel model. In this case the empirical model is usualy motivated by some sort of science, attempt to publish in the relevant scientific or applied math journals. Your primary contribution is evidence in the reasonable model.
As a synthesis of connections between many different areas of math that your conjecture brings together. Publish this as a survey that unifies the areas of interest. Your primary contribution is the connections between fields.


If your paper does not have any results apart from the conjecture, then such are usually disseminated informally. Tell your friends, colleagues, publish a blog post, ask for a proof on MO. This is a great way to find someone to work with in helping you turn your conjecture into a theorem.