QuickBooks Online is a great platform to house your business data with less paperwork, and I can appreciate also wanting downloaded copies for your records outside of the QuickBooks cloud. This ensures the data is on hand for you and that you can print or manipulate it as needed. I'll help you out with this.
When it comes to downloading or exporting data out of QuickBooks Online, there are a few key things that can be exported in bulk, which you can see here: Export reports, lists, and other data from QuickBooks Online. For things like invoices and statements, downloading them one-by-one is typically the way to go to handle this. For invoices, you can open an invoice, choose Print or Preview at the bottom of the transaction, and then download the transaction from the print/preview window's download button.
Another way you can go about this is using the Batch Actions feature on the Invoices page. Here's how.
This should open the selected invoices in one print/preview page, where you can then click download to have them all included at once. Keep in mind that this would put all invoices for every customer into one document. If you'd like it separated out by customer, go to the Customers tab instead, select the customer, check off the invoices, and then select Print transactions from the Batch Actions menu.
If by statements you mean customer statements, these can be downloaded by selecting the Print or Preview option from the Create Statements window.
Feel free to leave feedback for our product development team about any of these options by using the steps here: How do I submit feedback? This helps our product development team learn the ways that QuickBooks users would like to work with the program. Here's where you can stay up-to-date with new features: QuickBooks Online Features and Product Updates
Another option you can consider for downloading your data is to use a partner app. There are a variety of apps found in the Apps tab of QuickBooks Online that can help with tasks you're looking to complete in the program.
I know I've given you a lot of information here, so feel free to ask questions if you have any. I'm here and happy to clarify!