So many people submit applications in a day, that they probably won't even remember you. Usually I end up dressing up when I am called in for an interview.
During your job search wear what makes you feel comfortable just make sure you look tidy. If you want a better chance of getting the job, ask to hand in your application to the manager and give him/her a few words.
When gathering applications you can go casual...When turning in apps look a little better than casual (maybe a polo) because you never know you might be talking to the manager and you want to look at least presentable; I've also had friends who have been interviewed on the spot, but it's not likely to happen...And when you finally do get the interview; I always go with dress shirt, dress pants, dress socks, and dress shoes (I feel like tie is optional because it is only part time I'm assuming, but completely up to you!).