+1 to Brian's suggestion for you. the RM&C tool is powerful and might be a bit daunting, but once you set up a template you like, you can save it as a preset. So even if you use it for other tasks you can simply bring up your "mask" or template so you don't have to reconfigure it all the time.
Remember the expression "Garbage In - Garbage Out" ? Too much automatic kills any benefit -- why iTunes can make a total mess out of a library btw. So no the Rename Move tool is not automatic ... but if set up correctly ...
1) you have your dump down load folder as a watched folder for example thatyou import in JRiver
2) you open the tool Brian mentioned and you hit a button it will then do what you set it up to do ... rename the files in a coherent way, move them into another directory etc.
So it pretty much as automatic as you want it to be .. it gives you the option of an intial verification. Once set up, just click a button and done.
Now you mentioned downloading stuff, with legit downloads usually track number, Artist, Album, song name and Album Artist will already be in the metadata -- this is required with the possible exception of Album Artist maybe (itunes needs this too, but it tends to guess if it doesn't exist and it often guessing wrong). You might be part of a DJ Pool though which it is quite common where people's mixes and samples are shared (also legit btw ahem, not talking torrent here). Now the tags might not be set .. or might not be how they should be.
If this is the case, or even if you are buying flac recordings from Sites like Bandcamp or Qobuz, I find it easy, for me, to use an external free ware program (which can do what the RM&C tool can do, but also allows you before import to just verify and/or modify key tags, then it can automatically re-write the file names and/or automatically move the files into coherent directories. Its basically a free version of Tag & Rename ...
http://www.xdlab.ru/en/ its very good. You might not need or want this though, I'm just obsesive with my tags and prefer it to be at the top of my workflow. JRiver can also do an auto lookup for tag information (Libary Tools=Lookup Track Info from online database) -- once you let the Rename Move tool do its magic. Your choice.
A small learning curve, small price to pay for ultimate control
EDIT: I see this is posted in a version 18 forum ... this is old .. the Rename tool has been around awhile, but I'd highly recommedn buyin the upgrade. MAJOR cool features for audio in version 19 and 20 ... and if you change later you will have to reanalyze your entire collection if you ever use playlists/temporary volume leveling. You should do this before you set up any big collection. The upgrade price allows you to skip versions (ie go directly from 18 to 21). If you decide to do this, I recommend not to uninstall version 18 until your setup in v21 is working correctly