Keep in mind the rule changed recently and have tightened up. Getting more than $500 may be difficult on a decent card with no job (if that's the case). However, you should look at companies that your parents deal with that have cards. E.g., USAA insurance is pushing financial stuff for kids. For those that are with them (they are limited to membership -- but top Customer service surveys), you can apply on line and instantly get $500 card with good rates etc. That's what D did. There are other companies that do similar stuff, AAA does cards now. You really have to be careful however. For example because of her card she gets all sorts of pre-approved offers for larger credit amounts, but if you read the fine print they are really bad deals. And I tend to agree you probably want Visa or Master in terms of be taken most places -- though I have found Amex does a better job if you have a complaint -- but it can depend on the bank.
If the issue is to have credit, i.e., spend money you don't have it's one thing; but since you pay it off, another option is a prepaid card. Which gives you the flexibility of a card -- don't carry cash, with limited exposure. D has both, plus a debit card. Her limit will go to $1,000 shortly if we ask, but she does not use it that much.
Finally, a number of cards use rewards point which can go towards travel (airlines), but check out if there are annual costs. Also, to be realistic, a $500 or even $1500 limit will not allow you to easily accumulate much in the way of air travel in any reasonable time. I have one and put everything on it, and pay it off each month, still take time to get decent airline tickets.