Format the flash drive with a filesystem that both windows and Linux can read/write. The command 'mkfs.vfat -F 32 /dev/XYZ' (where XYZ is the usb stick when it's plugged in) would format the stick for fat32.
In Linux, you can mount the stick with 'mount -t vfat /dev/XYZ'. Write files to the stick using your favorite editor, and read with your file manager or editor apps, drag 'n' drop, etc. In windows, just click on the usb drive in windows explorer as usual to see files, drag 'n' drop, etc.