What to read

Something fun. It needs to be so much fun that you will look forward to reading it every day. It does not have to be intellectual, it does not have to improve your knowledge of science or history. Remember: you want to convince yourself that reading in English is fun. Don't feel guilty about reading comics, magazines, detective stories, romances, etc. (related article by Stephen Krashen)
Something challenging, but not too challenging. What does it mean? There should be some words that you don't know, because you want to learn something. However, there shouldn't be too many difficult words, because you don't want to use your dictionary 10 times in one sentence. There's a simple rule here: If you're not enjoying the text, switch to an easier one.
Something with the kind of sentences that you want to write or say yourself. Want to learn to talk about computers in English? Read an English-language forum on computers. When choosing a book, choose one with modern language and lots of dialogue. If you read a book written in obsolete English with lots of literary descriptions, you won't be able to use too many of these phrases in your own sentences (unless you write books in English). You want useful sentences that you can imitate.
Start by reading a few books by the same author (or a few books on the same subject). Each author has his/her own vocabulary and grammar. For example, when you read a book by Michael Crichton, you come across a lot of scientific vocabulary. Afterwards, it is easier for you to read another Michael Crichton novel than to read a book by a different author.
When you read another book by the same author, you will notice that you understand it much more easily than the previous one, and you will feel great about your progress in English. On the other hand, if you jump from author to author (or topic to topic), you will always be frustrated by unknown vocabulary and grammar, which is not healthy for your motivation. (related article by Stephen Krashen)

Some ideas

Here are some ideas of texts that you can read in English:

Literature. Whatever kind of books you like, you can read them in English.
Simplified books (e.g. the Penguin Readers series). These are popular books, re-written in simple language for English learners. They are just perfect for beginners and if I were to learn a new language, I would definitely use one of those. They are available in different levels of difficulty — the simplest ones use only 200 basic English words. Try the intermediate or advanced levels (over 1,000 words) — the lowest levels use so few words that they sound quite unnatural.
Science books. If you are interested in science, you can get great science books written in English. There are many famous English-speaking authors in many subjects, such as psychology, evolutionary biology, physics, or economics.
Textbooks. If you're studying at a college and you use textbooks written by English-speaking authors, you can get the original English versions. If you are learning a new computer language, you can use a book in English. You will learn your subject and English at the same time.
Forums and blogs. Forums, discussion groups and blogs are a unique source of written informal language. Unlike other written sources, such as books or newspapers, they are very close to the way native speakers talk. As such, they are an excellent source of input for English learners. Ideally, stick to forums for native speakers and remember that many native speakers make spelling mistakes.
E-mail. Like forums and blogs, e-mails from native speakers are a fantastic source of "everyday English", which is normally the kind of English you want to speak most of the time (except for some formal occasions). Communicating with a native speaker over e-mail gives you a lot of pleasure, as well as an opportunity to practice your writing skills.
Software. You can start using English versions of your operating system, your word processor, and other applications.