How to go about getting a "real" job?

It's probably going to be quite a bit longer than July 4th until until COVID-19 is a non-issue, and with that in mind it may be difficult to get these kinds of jobs because, depending on your state, you may be competing with adults who may be out of work due to the shutdowns. With regard to where you're working, you may know where you want to be employed, but that won't mean anything unless they're hiring. Usually you can find that information online, either for the store or establishment's website or the mall's website, if they have one. In any case, I agree with most of @BriannaM's advice.

To emphasize a couple of points, even though your personal email might be fine for getting this job, it'll be a good idea to make a professional email address that you can use for anything similar moving forward. Personal emails are usually fine for things like this in high school, but once you're in college employers, professors, etc. are going to start expecting a professional email that's just something like [firstname][lastname][birth year]@[gmail/outlook/etc.].com or something like that. Moreover, it's good to have a more professional address to give to anyone you're trying to network with—which is important no matter what kinds of jobs you're trying to get. It comes off better in the long run and you will need it eventually, so it's good to get in the habit now.

As for your resume, absolutely try to build one that includes a) schoolwork, current GPA, and relevant courses, b) clubs and extracurriculars that you're very involved in, and c) any commitments outside of school, like your current job. Being paid under-the-table doesn't matter for whether or not to include something on a resume, and you don't need to mention anything about how you were paid; just include the job on your resume with a brief description of your responsibilities.

Finally, you mentioned that you're worried about being "too professional" but, in reality, that's a pretty hard thing to do. It's like being "too polite"—not really a thing that's possible. Now, if you're worried that focusing too much on sounding "professional" might make you come off as stilted, then you might need more practice, and trying to do additional mock interview (if available) might be helpful. But otherwise, I wouldn't be too concerned about that.

Some tips that I can give you on your journey of getting a "real job" is to first of all acquire working papers (you should check if employment certificates or permits are required to work as a minor in your state). You should also create a high school resume, as it shows your seriousness in obtaining the job, highlights your abilities, education, and experiences, and should include it when submitting your application for employment. You should also secure recommendation letters from adults, teachers, counselors, etc. if they agree. You should also provide them with documents that may be needed, such as, your transcript, etc. Although, this doesn't seem important, you should use a professional email address when filling out job applications. You should also expect an interview and be prepared beforehand. You should do research on the company, practice answers to sample interview questions, dress appropriately, and bring materials that may be needed, such as, reference letters, your social security card, resume, or driver's license. Lastly, you should always follow up on your application. If you submitted an online application, you can call 7-10 days after submitting it to check on the status. If you meet with someone in person, you should send a thank you note.

There are several resources online that can give you information to several of your questions/concerns/wonders, here's a link to an article about building your resume:

https://www.ecampustours.com/for-students/career-exploration/building-your-resume/high-school-resume#.XsHAOGhKjIU

There are also many job search engines that could help you find jobs suited for you in your area.

I hoped this helped you, and answered your questions:)