The Gallery doesn't actually contain the files you're looking at. They are stored elsewhere and Gallery is simply caching those as thumbnails. If those files are in the cloud somewhere it's because you are using some other app that syncs them with your tablet. In short, they should already be off-line, somewhere on your tablet.
I don't have a Picasa app on my tablet. it's the Picasa website I go to on my computer. Do you mean to download them, from there, to my computer -- then transfer them to the tablet with the cable?
Or can you recommend a good photo app, for the tablet, that would do that for me?
You should be able to use a browser on your tablet to log into the Picassa account. Despite not having the app, you must have added it as an account. With that said, yes, you could download them to the PC then copy them to the tablet.