Alright, I am guessing I am probably older than the average person that might be posting here, so, for me, when I hear “older anime” I think back to shows/movies pre-2000. Limiting myself to that criteria, here are some that come to mind (in chronological order):

• Grave of the Fireflies (1988) - If you have never seen this movie, then you should. Just prepare yourself to be an emotional wreck for a good while afterward. My first introduction to this movie was in 2001. Back then, on September 11, the school was gathered into the main auditorium after the planes struck the towers. For some reason, some teacher thought it was a good idea to show a movie to the students to keep them entertained. I never found out which teacher chose it, but somebody decided Grave of the Fireflies was the movie of choice for the moment. Yeah… It has made for an interesting anecdote at parties ever since.
• My Neighbor Totoro (1988) - This probably remains the easiest entry point into anime for non-anime watchers that exists. There are many that argue other Ghibli movies might be better in one respect or another, but I think this one is special due to how approachable it is for just about any potential viewer regardless of age/gender/previous anime exposure.
• Yu Yu Hakusho (1995) - In my opinion, this series took a lot of the shonen conventions established by its predecessors like Dragonball (1986), and refined them. I am not about to claim it is the best shonen, but it established a lot of the best practices so to speak for what makes a good shonen and its influence can still be seen in shows today.
• Cowboy Bebop (1998) - This one is a classic by just about any definition. It is still talked about regularly in forums like this one. If that isn’t still relevant, then I don’t know what is.