Even Chromebooks have a Downloads folder, which is the default saving location for downloads and screenshots. Downloads can be accessed via the File manager app (blue folder icon in your Apps) but here's a setting that everyone should apply:
In Chrome settings, scroll down to "Show advanced settings"
Scroll down more to find the Downloads section
Check the box for "Always ask"
With this box checked you will now be prompted to choose a location (eg. save directly to Drive) and (perhaps more useful) to rename the file before saving it. This can save you a lot of time when downloading images, PDFs, etc!