Can I choose the order products appear in on my category pages?

Not by hand yet. There is no drag-and-drop or position number for products in a category. By default, listings show the newest products first.

Shoppers can reorder a listing with the Sort by dropdown. Out of the box it offers Price: Low to High, Price: High to Low, Name: A–Z, Name: Z–A and Newest First.

You can rename the dropdown and its choices, or switch choices off. Go to Settings → Appearance → Catalog Settings → Filters and edit the Sort row under Sort & Filter Types. You can't add new sort choices there.