Present math for non-mathematicians
I would recommend blogging. This seems like the standard procedure for explaining your papers both to the lay audience and to other mathematicians (potentially ones that don't work on your specific problem). The blog format is preferable over arXiv or journal publications for reaching the lay audience because it is more accessible. Links to blogs are easier to share and faster to read than journals (which might be behind paywall!) or pdfs from arXiv. A blog setting also allows you to interact with your audience through the comments, this is the best way to help guide them through any confusion.

A blog setting can also be used to provide casual tours through proofs for experts and graduate students. This has recently started in TCS. Since this is aimed at the slightly technical audience, it is more appropriate to put on the ArXiv, although I would still advocate blogs.