Accepted a low salary, how to bring it up again?

Telling them you got another job offer would be risky.  They might tell you to take it.  That aside, why should they pay for your gas in getting there?  That expectation is ridiculous.  If you express the audacity that you have described, expect them to tell you to go and find another job, because they will tell you that you are fired even before you start.  Accept the money you AGREED TO.  In one year, if they don't give you a raise that is acceptable, THEN ask for a little more.  Pay for your own gas.

You mom is wrong.  For a 19 year old, $15/hour is not a low salary.

You are 19

You don't have a degree

You have no real experience in the field

You didn't give a low ball amount, because you have no history to support anything you ask

If they officially invite you to the office, they may or may not office a mileage reimbursement.

- remember, that almost every candidate they look at is a better candidate in most people's eyes than you.  They are not 19 and this is not likely their first job ever held "outside the family business".

- if you want to ask them for mileage reimbursement, you can.  You have to be prepared that you may lose the opportunity over a few gallons of gas.

If the company is a legitimate company, then they know what the position is worth too them.  At this point in your career, you will always get the bottom of the range.  But they will pay you the bottom of the range, not what you asked.

"freelance designer" is this a freelance position?

If it is, you need to do research.  With 15.3% SE tax, I don't see how anyone can work freelance for less than $25.  That doesn't mean you should be earning $25, it means you should be working working in a company as an employee, if you cannot command $25.

You agreed on a wage and accepted it. End of story. You could try your little ruse about another job offer, but that is likely to back fire on you. Then what? Two hours away is nothing, they should not be expected to cover your travel. If you had to buy a plane ticket, book a hotel - then sure, they should cover that. You have a lot to learn about the working world. You are working for them, not the other way around.

They are not going to give you more after you just accepted $15. Prove yourself for 3 months then ask for more. I worked for minimum wage at your age.