How do I set up the component so that the entries can only be edited by the user who made the entry. Admin user would have full access to all entries.

You can do this with the help of your ACL's
Make your user Author and give him EDIT OWN privileges. And it will have your setup ready.

This is very old topic... lol

The answer is correct, but incomplete.
If you want that the others users do not have access, then you must combine with published = 0
(Published wizard + default value:0)

Then when an item is published, the author will still have the only access to edit, and the item will be shared.
Very simple indeed.

Thanks for the rectification admin, you are quick.