Edith Hamilton's "Mythology" is always a good place to start for Greek and Roman, and a little Norse. Others may be able to recommend for other cultures, my main experience is with Greece and Rome.

I'd recommend "The Hero with a Thousand Faces" by Joseph Campbell. It is from 1949, and there are certainly many critiques on both structuralism in general and Cambells' monomyth in particular. But, seeking commonalities in myth and hero stories he covers a lot of them, the book is beautifully written and much of later fiction is explicitly modelled after Campbell. It is one of my favourite books, I've recommended it before to people with general interest in mythology and they also found it very interesting. In any case, it is an excellent read and you'll likely remember it often when reading other myths, or even just watching films or other fiction.

There are several books of Irish mythology, the Tain being one of the more famous, which were written by monks in the 12th century. These are not books about Irish mythology, they are the books OF Irish mythology, or as close as you can get. Some Christian spin laid over the stories but they're wonderful nonetheless! Fascinating stuff, and relatively easy to read if you get a good copy.

I recommend you listen to The Teaching Company's lecture series on mythology by Dr. Elizabeth Vandiver. It's great.