(This is a Post-In-Progress: Comment with questions here and I'll expand the post accordingly)
First go to this page, download all the tools, and put them in the same folder.
Next, download this zip file. It contains a set of open-source code and your compiler
This video introduces the $10 computer's graphics system—it is a great place to start.
This video introduces the YY-CHR graphics editor. If you want to do some game hacking right away, here's how to get your hands dirty:
First, open the ROM, sprite.nes (found in the zip file) with one of the emulators, Nintendulator or Nestopia. See what is going on here? This is the open-source code you'll be altering. Explore the menus of Nintendulator and Nestopia, as they will let you see some pretty interesting things about the ROM.
Now, open the sprite.nes with CHR-YY and try modifying the tiles. Save the file and re-run the .nes file in the emulators. Cool!
Now, open the .nam file with name.exe (in the tools you downloaded). You should try to alter the background, and resave. You'll need to run compile.bat after this, before you run sprite.nes.
That's the first activity! Any questions?