Some notable films you might want to check out (but might not realize are worth it)
Repo Man 1984 -- a major role for him in a cult classic from the punk rock era. Also young emilio estivez
Paris Texas 1984 -- A great film from the german director Wim Wenders, very much like a german David Lynch
Pretty In Pink 1986 -- A John Hughes film, Harry Dean Stanton plays Molly Ringwald's father in a touching performance
Wild at Heart 1990 -- An avant garde film by David Lynch with Nicholas Cage and Laura Dern in their prime. Probably Lynch's best movie, in my opinion.
The Straight Story 1999 -- Also by David Lynch this is actually a G rated film, about a guy who goes to visit his brother. It's kinda spectacularly touching while still being extremely minimalist in story.
Big Love 2006-2010 -- HBO TV Series about a mormon polygamist marriage, where Harry Dean Stanton plays an evil cult leader, super creepy, super effective and a "you love to hate him situation".
That may be a bit too many suggestions, but I pared it down from the 199(!) credits he has. I have seen all of the above and can recommend them all, though several are avant garde or not designed to be happy fun movies.
Also note he has a movie coming out soon-- Lucky -- where he plays the titular character and it looks to basically be an ode to Harry Dean Stanton.
RIP Harry-- you impacted my life and I will miss you.