You need to not only set the owner and permissions on the directory where you mount the drive, but you also need to set mounting options in fstab.
I use the preferred method of UUIDs rather than your older method of /dev/sda. If you want to use this, it tells you how at the top of fstab, but at a terminal, you enter the command
sudo blkid to get a list of UUIDs, which you can then cut and paste into fstab.
My NTFS drive is set with these options; I am the owner:
#sdax good idea to add comment here
UUID=C61460BE1460B2DB /mnt/foldername ntfs-3g auto,users,uid=yourID,gid=yourID,utf8,dmask=002,fmask=113 0 0
I mount it in /mnt/foldername, and yourID should be changed to your login name. This should get you started, but if it isn't quite right, you can either learn more by searching, or by asking here.
I don't actually have any FAT/FAT32/VFAT partition, but I think they may be the same, except for the ntfs-3g part.
the /media directory is traditionally used for auto-mounted media, I think, but I suppose it's OK. I use /mnt.
Edit - testing mount options
One easy method for testing and debugging the options in /etc/fstab is to use a special mount command that will mount all the usual mounts in fstab interactively, rather than at boot time.
My method is to keep open the file
/etc/fstab, but save any edits; then, at a command prompt, I enter
sudo mount -a to mount everything in fstab, and it will output an error message if there are any problems. If there are no problems, I test to see if the results are what I want. If not, I unmount that directory only, using
sudo umount /path/mountname, make another edit to fstab, save, and repeat until it's working.
If there is a syntax error, then before closing the file, I usually comment out the entry if I can't get it fixed, and try again later.