Which English classic should you start with at B1?
Start with a short, plot-driven classic written after about 1850 in plain English: Alice's Adventures in Wonderland, The Adventures of Sherlock Holmes, The Great Gatsby, Tom Sawyer or Peter Pan. Leave Dickens, Austen and anything in verse for B2.
What B1 can and cannot do
At B1 you follow the main line of a story, you understand dialogue, and you guess a fair share of unknown words from context. What you cannot yet do is hold a sentence of sixty words with three subordinate clauses in your head, or read irony that depends on register. So the first classic should be one whose difficulty is in its vocabulary, which a translation covers, rather than in its sentences, which nothing covers.
Five that work
Alice's Adventures in Wonderland: short chapters, short sentences, a story you know. The wordplay is the hard part, and the translation shows you how a translator solved it.
The Adventures of Sherlock Holmes: twelve separate stories, so you get twelve endings. The vocabulary of crime, rooms and weather repeats from story to story, which is exactly what a learner wants.
The Great Gatsby: about 48,000 words of clean twentieth-century prose, the shortest of the great American novels. The first chapter is the hardest; after that it moves.
The Adventures of Tom Sawyer: plain narrative, a lot of dialogue, some dialect you can safely skim.
Peter Pan: written for children but not childish, with sentences that are long and yet easy because they are built from simple parts.
What to postpone
Dickens and Austen are B2 books. Their sentences are long, the humour is in the phrasing, and a B1 reader spends the whole time in the translation. Poetry, Shakespeare and anything translated from Greek belong to C1, even when the story is familiar. They will still be there next year.
The B1 shelf holds every book in the library rated at that level, and the B2 shelf is the next step once a B1 book stops needing the translation on most lines.
How to tell if the level is right
Read one page without the translation. If you understood what happened, even with several unknown words, the level is right. If you understood the words but not what happened, the sentences are too long for now; drop one level. If you needed the translation on every line, the book is not wrong, it is early; put it on your list and come back.