I never truely counted Speed Racer as anime, just a regular toon. It never really felt liek it was anime to me. Either way, I guess my first would have been Macross (Robotech in the US and thus Robotech when Australia recieved it). Very cool show but back then, I didn't really associate it with anime. The first stand alone anime movie that I saw was 'Akira' and to this day, I still carry the scars. I really can't stand that movie. Still, next in line was 'Ghost in the Shell' and since then I haven't looked back. Titles that followed were things like 'The Castle of Cagliostro', 'Riding Bean' and the wonderful 'You're Under Arrest' series. 'Project A-Ko' was in there somewhere too.
First anime has to be something like card captor and Dragonball, then dragonball Z, from that i moved on to Neon Genesis Evangeileon, full metal panic, full metal alchemist, FLCL, Hellsing and not forgetting ghost in the sheel 1st gig and film