If my child is under 24, goes to school full time and has a part time job, can I claim them as a dependent, and do they still have to file their own taxes?
Yes if they are under 19 or under 24 and a full time student you can still claim them.
You don't include their income on your return. It would only go on their return if they are required to file one. To file a separate return for your dependent you need to set up a new account. Online is only good for one return per account. You can use the same email address for 5 accounts. They can probably use the Free Edition. How to start a new online return https://ttlc.intuit.com/community/using-turbotax/help/how-do-i-start-another-return-in-turbotax-onli...
Be sure on their return they check the box that says they can be claimed on someone else’s return. If their only income is W2 and under 14,600 they do not have to file a return except to get back any withholding taken out.