DMy days off are irregular.

Since it's a title, it's not strictly necessary, but I still feel like it'd sound more natural if you added an article.

I'm working the next weekend.

Usually it's just said without "the" when it's referring to the upcoming weekend. "The next weekend" usually is in comparison to another event you're talking about. E.g., "I'm going out of town next Thursday, but I'm coming back the next weekend."

Alternatively, it could also be "this weekend."

I wasn't sure how to write "不定期" things in English, so I googled it.

"google" is a transitive verb, meaning that it has to have an object.

And I found an expression usingthe word "irregular."

Your sentence isn't grammatically incorrect per se, but I don't think it's what you were trying to convey.

I changed "expression" because it's just a single word, but also not totally necessary.

Stylistically, I would also just combine this with the previous sentence because it's so short. It would then be "I wasn't sure how to write 不定期 in English, so I googled it and found the word "irregular." (I cut out the "I" in this example because the combined sentence would otherwise have redundancy.)

( I wondered which adverb I should use in this sentence: occasionally, sometimes, often, etc. / Of course, I googled. it.)

"/" are not typically used this way in English. It kind of means "or" or "and" most of the time. Just separating with the period works; otherwise a semicolon or em dash could as well. (And also added the object to "googled" again.)

I don't haveSometimes, I have neither Saturday off nor Sunday off sometimes.

Or: "I have to work both on Saturday and on Sunday."

I wasn't sure how to write "不定期" things in English, so I googled.

I liked it because "regular" is one of thea familiar words for to me.