Should I wait to apply for a credit card?

I'm 20 years old and I'll be going into my third year at a public university in California. I haven't taken out any student loans or other loans of any kind. I have a debit and savings account through Bank of America and a brokerage account at Charles Schwab that my parents invested in before I was born so that I could pay for college. I've been working one job on campus for the past year that I make around $200-$300 per month. I also receive oil royalty checks every month that vary between $200-$400 depending on the price of oil. I haven't been working over the summer but in the fall I'll be working two jobs on campus and (hopefully) be making more money. I'm leaning towards applying for Discover. I want to build my credit by using it for things like gas and groceries. I also might co-sign on a used car soon with my dad, who has great credit. Should I wait until the fall semester starts and I have recent and updated proof of income to apply for a credit card? I don't know how many hours I'm going to be working, but the maximum I'm allowed to work for both jobs combined is 20 hours a week. I've also had straight A's my entire college career, if that will help with the process.