Give this a try..
* Stop the Transmission service
* SSH to your NAS as root
* cd /volumeX/@appstore/transmission/var # where X is which volume it's on, often 1
* vi settings.json
* modify the download-dir line to change the download destination. Alternatively, you can break the downloads into complete and incomplete by setting incomplete-dir to where you want incomplete torrents to go, and incomplete-dir-enabled to true.
* setting the watch-dir directory will let you upload to a particular directory (typically 'incoming') and transmission will scan that dir for new .torrent files, and add them when they appear.
* save the settings.json file
* make sure the directories you specify exist, and are read/write capable by user transmission
* Run the Transmission service
Synology uses 'busybox', which saves space but has a pretty reduced commandset, and it looks like the only editor you get is vi, so if you don't know vi it'll be a good opportunity to learn!