GY started a recommendations thread way back on OTF. I'm not an avid follower, but there are some really good films out.
The BFI on the Southbank has a twice yearly review of the best Anime from the last few years, and usually shows around 4 - 6 films, interspersed with some classics. They're showing Akira in a couple of weeks. I noted that the Prince Charles also puts this on sometimes.
One that I've seen recently and would thoroughly recommend is Paprika (wiki page contains major spoilers)
It was a major influence on Inception, which I've not seen, it can be slightly confusing to follow (but nowhere near the extent that some reviews make it out to be nonsensical). The main idea is around the development of new technology which allows people to view others' dreams and take part in them to some degree to assist with mental health problems. Things go awry when the prototype technology is stolen and allows the thieves to access others' dreams for their own purposes. The way in which dreams are portrayed are quite clever, totally bonkers of course, and the eroding defining lines between reality and dreams is quite unsettling.
I would recommend Five Centimetres Per Second, Summer Wars and The Girl Who Leapt Through Time. I'm not massively into it, but one of my best friends is obsessed with it and I'm still dipping my toe in the water, as it were.
I'm pretty out of the loop anime-wise, as I've not watched a lot of it since my teenage years. Looking at the old thread there are some good shouts, Perfect Blue being a definite recommendation if you liked Paprika, Ivan. With Paprika, someone bought me that on DVD, and I know what you mean around the confusing nature of it, but it's good fun and pretty creative. There's the usual big suspects, Fist of the North Star, Akira, Ghost in the Shell, etc etc, which are of course all worth a watch. And there's some bonkers stuff floating round from the early '90s or so, like Wind of Amnesia, which is an interesting premise, but ends up going a bit weird at the end.
I've got affection for the likes of Appleseed and Macross Plus, although objectively, neither are particularly stand-out, but I'll always recommend Cyber City OEDO, which I bloody loved and no-one else seemed to.
I used to have some cracking stuff on VHS back in the day, like the full set of Doomed Megalopolis, but in a fit of misguided generosity (and since I'd binned by broken down VHS player) I gave a whole pile of the things away.