You might consider the Discovier IT for students as well. However, I think that your best option would be to open a credit union account to deposit your paycheck into this summer. You can then build the relationship with them and apply for their in-house credit card.
It is great that you're here at such a young age. Most of us have been working on this for years, if not decades; the cards you see us applying for take years of management to get. That said, you don't necessarily have to start at the very bottom, with a secured card or a terrible lender like Credit One. Credit unions can provide a good way to get your foot in the door and start your credit file. In a year or two, you might be able to reconsider what cards to apply for.
The $3500 you'll make this summer will help a lot, so long as you get paystubs. I wouldn't expect very large limits, but you don't need them yet. Even with the credit union, I'm guessing that you get a limit of $500. Build your assets and income, show an ability to pay your card, and I'm sure that the credit union will increase your limits in no time.
