Linux does support FAT32 so you should have no problems using your flashdrive. Is the drive mounted before you try transfering data?
You can check if your drive is mounted by typing the following in the console after inserting the flashdisk:
The above will show you the mounted partitions and the filesystem type. Look for fat or fat32 or vfat
You could also type the following:
either one will show you if the flashdisk is mounted and where it is mounted.
In case it is not. look at the end of dmesg for the flashdisk and mount it yourself. Do this right after inserting the flashdisk
you should see something like sdb1
Now mount it:
# mount -t vfat /dev/sdb1 /mnt
Now you can go to /mnt and find your data.
You can also try and install samba and get a windows share on your linux.
Best of luck.