I also love classics, but I read modern books too. If you are determined to read classics, you could try reading something that you've read before. Right now I'm reading Oliver Twist, and as you well mentioned has a ton of complicated words.
Depending on what you like, I have the next recomentations:
First of all, I believe modern books are the best to get you into a good reading pace, once you've read a couple you'll be able to move on into more complex books without suffering.
For romantic cheesy novels, when have Danielle Steel and Nicholas Sparks. They have like a million novels for you to choose.
I also started reading all of the typical novels for young people like Narnia, The Hunger Games, Divergent etc.
Then, for the slightly less modern and a little more complex:
For Sci-Fi we have Farenheit 451 from Ray Bradbury, 1984 from George Orwell and I also loved The Ender Games, oh and The Martian!
For kinda historical fiction, we have Ann Of Green Gables, and The Books Thief (that one is modern, but not the typical american language)
You could also read stuff from other parts of the world so you get used to slang, there are great books about the slavery times in the us, or about scotland with tons of weird words for you to learn.
Hope this helps, and let me know if you would like to know about any specific genre.
Oh, right, I'm also not English native and love to read haha