Open the order you want to update and click on invoice button.
Go to the bottom right of the page and select 'Capture Online' or 'Capture Offline'. If you select 'Not Capture' The order status will stay as processing.
After you created the invoice, click on ship button once you shipped the order.
Enter tracking record, tick checkboxes to email copy of shipment and append comments.
If you see order status as Processing even after you shipped order. Go to tab 'invoices'.
Click the invoice to open it
Look for button capture on top right corner and click on it.
After invoice is captured, the order status will change to 'Complete'
I'm new to Magento and still have many questions. You're writing how to change order status to Complete, but what if I need more statuses (like pre-order, shipped, etc.)? Is it better to use an extension in that case (I found order status by Amasty)? Or is it ok to make such changes programmatically?
Thanks in advance