ConsulCaesar: The problem for me with older games that I didn't play back then is not old graphics as much as a dated interface. So I'd recommend you start with games with a simple but practical interface, such as post-Monkey Island point-and-cick adventure games.
Indiana Jones and the Fate of Atlantis holds up extremely well, including its pixel art. I would even say
Monkey Island itself, but it's been remade so the graphics and interface are different (although you can choose to play with the classic look and feel).
Absolutely have to second this. As it turns out some things about old games don't translate too well. Resolution, UI design, etc.
Some additions to the thread:
Tyrian 2000 I don't know exactly why, but it doesn't really feel dated and just works out. Everything is more or less intuitive. It's also free, so you're not risking anything taking a shot here.
Master of Magic Demonstrating how far game design has come, there's one simple reason why I'd recommend this: tooltips. Rightclicking anything brings up a small textbox explaining what it is. While it's still a good idea to read the manual, this is
massive. Really easy to get into, if you are roughly familiar with how civilization plays.
Both games are great, cheap, and utilize DosBox, so you can still fiddle with the config a bit.
Also
Albion.
While I can't think of a good JRPG for PC off the top of my head, there are similarities here. Premade characters with stronger integration into the story for example. The game doesn't necessarilly do anything special that would make it a good starting point, but it's not too difficult to get into, either. Generally aged well with the exception of the first person sections. It's also one of the best RPGs ever made, so that helps.