Congrats on getting the job! I know from personal experience that it can be hard to get them, especially if you need to work part time.
So I have a couple of tips for the future and for anyone else. First off (was going to put this at the end but its more relevant since you got a job), if you absolutely hate your job and want to quit, don't. Reason being is that while searching for another job employers will find it better that you actually have a job as opposed to being unemployed. On top of this take the job you dislike seriously, work hard and get a good recommendation off of them. When you go in for another interview you can list them as a reference and the person who is interviewing will see that even if you dislike something you still work hard. Also, you're making money while still searching for the next position (which as you know might be a while).
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Okay, on to applying. First off this is for lower starting jobs before you get your college degree, the tips above by teal6 are good for after you get your college degree. Realize that you don't have a lot of work experience and so your resume is going to look a bit skimpy to start off with. My resume to start off with was barely a page and I was really struggling to put stuff on it. My dad's work resume is about 10 pages long and is really impressive. There is time to fill it out so don't despair. Most people just open up a word document, but one thing I found really handy is searching resume templates on google.
https://drive.google.com/templates?sort=user&view=public This is really good because you can choose a style that works and they will have sample info that you can change to suite you. They look better than just putting stuff on a word document. As far as stuff to put on goes, put on every "job" you can remember throughout High School. When I mean job if you went over to your neighbors and did yardwork for them over the weekend put that on as you can then use them as a reference. In the future you can replace it with actual work experience, but in the beginning you want to get any decent amount of work you have on there so the interviewer can see that you are actively going about working. You also want to put on any notable accomplishments you've had at school. All of this is to help paint a better picture of what kind of worker you might be.
References, references, references. These are super important! Its a way for your employer to check your work ethic that isn't from your mouth, so something more concrete. This is why even if you hate a job you should work hard, you can think that you're working for that reference. To start off you can list stuff like coaches and teachers, make sure you ask them first. Some people don't like being references.
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Next is putting out resumes / applications, you did a bunch of things right. You put out a ton of applications, you followed up in the store, and you switched from online applications to paper. All of these things tend to increase success. The only thing I would add on is that you should ask to speak to a manager when you go in for applications. Just tell them, "Hey just stopping in to say im interested in the job and just wanted to put a face to the application". This will show initiative and when the manager is going through applications they'll be like, "Oh yah I remember that guy."
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Finally if you get called in for an interview there are a couple of things that you should do ahead of time. Clean yourself up. This includes getting a haircut if your hair is messy or unpresentable. If you keep your hair looking decent make sure that you comb it before the interview. Shave off any facial hair, the exception to this is if you can grow a thick nice looking beard. If you can make sure to trim it up. If you have any tattoos make sure that you can cover them with clothing. Take out any piercings that aren't in the ear. Dressing for the interview is important, at the very least wear nice pants and a shirt and tie. For women you can wear a nice dress or business dress as an alternative. It really depends on the interview, for a restaurant gig or a more manual labor job I would wear the shirt and tie, but I've gone for a couple of sales jobs and have worn a full suit (might of been overkill haha). Before you go into the overview make sure you have a copy of your resume. When you go in you can hand it to them again incase they have lost it. This also gives them something to look at so they have a quick idea of who it is they're interviewing.
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When you walk in, introduce yourself and shake there hand. Hand them your resume and then when they ask you to sit, sit. During the interview you should always look them in the eye while they are addressing you or while you are addressing them. Make sure to smile while answering questions. Be open and give more detail than just yes or no questions. If you are going to an interview where you'll be required to display some skill make sure you prepared and practiced ahead of time so when you do it, it feels natural. Make sure that you are able to give them an exact availability, like if you need Saturday's off for something or if you have school. From the get go they will be able to plan accordingly if they do hire you. I've had problems in the past where employers have had trouble scheduling me in relation to school and if you give your availability right off the bat nothing can fall back on you. Also know when you can start, most places will ask for a start date (I usually say asap).
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Finally some closing thoughts, just remember that the job you land won't be the one you'll have forever. You can always get another, but treat your job as something you are getting graded at. If you work hard you will get promotions and raises and when you move on you'll have a good reference for your future job finding.
Another way to go about getting a job is through friends of your parents. Tell your parents to mention to their friends that you're looking for work, I've gotten two jobs in the past because my dad mentioned that I was looking for work and they offered me a job. Your parents can be your advocates when you're first starting out.