Regardless is a legitimate word that every single person on earth acknowledges as a valid word.
Irregardless, on the other hand, is an informal variant of regardless. It can be viewed as the illegitimate child of irrespective and regardless.
If you use irregardless (especially in your writing), you will incur the wrath of many people who will be more than happy to point out that irregardless is an “uneducated” word. (Some people even put irregardless on the same level of formality and legitimacy as ain’t, i.e., not very high.)
So again, the answer to your question is a resounding
Simply use regardless or irrespective, look smart, and move on.