Tinkercad does not automatically save your work through the design process, so it is important for you to explicitly do so. The save button is found in the Design drop down menu at the top-left corner of the screen.
When your design is ready for printing or sharing with others outside of Tinkercad there are a few options found in the Design drop down menu at the top-left corner of the screen.
To create a standalone file of your design choose Download for 3D Printing. The screenshot below shows the available file format options. If printing in HSHSL's Innovation Space you'll want to choose the .STL format. In fact, the .STL format is the most commonly used format for 3D printing.