I make $13 an hour paid weekly. My check be $420. What is a good amount to start saving up for my own place?

All of it. Houses cost a lot… An apartment might be the best bet, but you still might need a roommate to help afford rent and necessities.
Depending on state, $13/hr, and $420/wk is not enough of an income to afford a house… An apartment, maybe. However, you need to start saving and cut down on spending things that are non essentials. Cut down on going out to eat, buying clothes, etc.
You still have a long way to go before you are able to afford a house and be able to live comfortably - not paycheck to paycheck.
$13 an hour is chump change that can barely make a decent living, yes you can get yourself a low end cheap apartment. Don't expect much with making $13 an hour.
Get a single. And save 1/4 of your pay check. And try to get at least 17 a hour in the next 4 months
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2Opinion
That is a good amount. If you can't save now, you never will
All of it if you live at home
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