You don't need Picasa, or any other photo viewing software to do what you are wanting to do.
The way I would do it is to make sure that the memory stick is empty, or use an SD card - also empty. This will show up on My Computer as a Removable Drive with a drive letter of F, G, H, etc, depending on how many USB ports your computer has. Double click on this drive letter and restore it down to a size of approximately half the size of your screen.
Then open the folder(s) that contain the photos you want to put on your memory stick. Select all the photos you want by left clicking and holding Ctrl key down. When you have selected all the photos you want, just copy and paste them across to your new removable drive.