How long should I stay at a retail job before I start applying to a new one? Or should I just not bother?

I've been working as a cashier at a grocery store, for about a month now. It's my first job so I'm glad I'm getting some work experience under my belt, It's not awful, the people I work with are nice although I don't know them that well, but it's safe to say I'm not exactly a fan. It's not that the work is unpleasant it's simply that it's tedious dull and boring and the shifts can start really early or end really late. The store is also pretty understaffed. I've been thinking about finding another job, but my options are still probably going to still be retail or restaurant jobs. There are a few jobs in my area that I think have the potential to at least be a small improvement over the job I currently have; either because I like the store/company better or the hours seem better than the ones I have. My current plan is to work at this job for another month or so and then start applying to jobs. Should I wait more time? Less time? Or is there little point in doing this at all?

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