Hello there!
Well, I daresay you've come to the right place if you're looking to expand your anime experience
A good place to start would be to start looking into famous/popular anime that have aired over the last few years or so, seeing what kind of genres you generally like watching, and go from there. Studio Ghibli was also a starting point for me (although I was very young at the time and didn't realise it); but over the years, whilst their anime films are very high quality and interesting watches, the stuff they have is just the tip of the anime iceberg. Another small distinction would be that most anime are usually formatted into 25 minute long tv episodes, instead of 1-2hr full length films.
Some of the titles I would recommend to watch off first would be some golden oldies like Neon Genesis Evangelion and Cowboy Bebop to relatively more 'recent' classics like Tengen Toppa Gurren Lagann, Death Note and Detroit Metal City. A lot of people would probably disagree since it all depends on personal preferences.
Personally though, one series that I would highly recommend to you would be Code Geass - awesome story, interesting and well developed characters, as well as other stuff.
Plus it's about an alternate future British Empire ruling over North America and most of the world. Here's a generalised chart/graph thingy I found somewhere...
