My joint favourite is The Last Chronicle of Barsetshire, but you can't read that without first reading the previous five. The first two are The Warden and Barchester Towers, which various people have already mentioned. Trollope has a wonderfully accessible style, I think. And a good place to start for a novice 19th century novel reader. If you find you like the Barsetshire novels there are many more of his novels to recommend, including the six Palliser novels.
George Eliot: Middlemarch and Silas Marner are great, but I wouldn't start with either. Middlemarch is off-puttingly long and Silas Marner takes ages to get going. Adam Bede and The Mill on the Floss are better ones to start with.
Dickens: I love them all except The Pickwick Papers. My favourites are Bleak House and A Tale of Two Cities, but I'd start with Oliver Twist, Great Expectations, David Copperfield, or Nicholas Nickleby.
The Brontës: I confess that I don't like Wuthering Heights at all, and I'm not really a fan of Jane Eyre either. I'd start with The Tenant of Wildfell Hall.
Wilkie Collins: The Moonstone and The Woman in White are both great. I'd start with the latter.
Hardy: I love them all, but would start with Far From the Madding Crowd, Tess of the D'Urbervilles, Under the Greenwood Tree or The Mayor of Casterbridge. Stay away from Jude The Obscure until you're sure you like Hardy!
Jane Austen: again I love them all. My favourite is Persuasion (my joint favourite novel of all time), but you could start with any except Mansfield Park.
European novelists: I wouldn't read any until you have a good few English novels under your belt. In particular I'd stay away from Crime and Punishment and Les Misérables for the time being. They're great reads though. I'd also leave Anna Karenina and War and Peace until later. Ditto Madame Bovary.
More modern classics: I second people's recommendations of: Rebecca, A Suitable Boy, and especially Captain Correlli's Mandolin.
I also love Midnight's Children by Salman Rushdie, the 'This Dark Night' trilogy and The Return of the Soldier, both by Rebecca West, The Call of the Wild by Jack London, My Family and Other Animals (and sequels) by Gerald Durrell and I have a real soft spot for Watership Down. In fact I love a lot of children's books, but that's a whole other thread.
Not at home at the moment, so can't cast a quick eye over my bookshelves to see what I've missed. There will be many.
Happy reading!