there are two kinds, one is arthouse, that's with unconventional multilayered plots with so much symbolism and the other is movies with some artistic touch...sometimes its hard to tell one from the other I'll suggest you a movie that many of us first saw in the arthouse area...mulholland dr..prolly you've heard the name. watch it. then go for movies like the thin red line, apocalypse now
arthouse movies were mostly the movies from europe, hollywood don't really make them, at least to my understanding. so I strong suggest david lynch for you to start with. if you want downright arthouse, check out movies like, solaris (70s russian movie) this is a good starter.