How to export a Google Drive document in a different format:
Open the document you want to convert.
Click the "File" menu and select "Download as." A list with a variety of file formats is presented. Pick the format you'd like to export as.
This will download the file to your computer.
How to convert an uploaded document to Drive-friendly formats:
After uploading your document to Google Drive, right-click the document and select "Open with" and choose the proper format to convert to:
Word (.docx) files -> Google Docs
Excel (.xlsx) files -> Google Sheets
PowerPoint (.pptx) files -> Google Slides
In this example, I've converted my test Word document to the Google Docs format.
You will then have two nearly identical files in your Google Drive. You can differentiate between them by their icon:
It is safe to delete your initial Word document. You can then share and collaborate simultaneously on the Google Doc file.