What is the best method/process to learn an accent? For example, Irish.

The best way is to immerse yourself with people who speak with that particular accent. You will find yourself starting to speak with different a different inflection and accent.

I went to Israel over the summer for about a month and found it tough to shake myself of the Israeli accent for a few weeks.

Lacking the ability to immerse yourself, listening to it spoken (audio) should help.

Movies movies movies! I recommend The Secret of Roan Inish and Waking Ned Devine to start.

For some people, they are just more easily affected, so an accent would rub of on them, but others, we need a method, a process.

Does anyone here have experience in acting, accents, or phonetics?

There is only one real way to learn Irish accents, and that is to frequent Irish pubs.