I originally started with web resources, but here's the stuff I credit with really advancing me properly, roughly in the order I read or used it:

1. TTMIK's Hangeul Master. It's really easy to pick up hangul basics so spending $20 on a book that's superficially just about the alphabet might seem silly. But the harder part to learn isn't really the symbols, it's the pronunciation. You need to get to grips with the various sound change rules that apply when various characters are adjacent to each other, and with the pronunciation differences between the normal and tense consonants, and so on. If you can't look at a word and confidently form its sound in your head you can't read or write efficiently, so this is crucial. TTMIK's book is really a solid package: You get the basics (the symbols, the stroke order, etc.) but also a good introduction to the sound change rules, not just on paper but also in really well-done supplementary audio resources and quizzes (where many books fail). Finally there's a second section on various Korean handwriting styles that I found surprisingly useful (Korean TV shows and other materials often use pen scripts that can be hard to read without spending some time with this).

2. To complement the above, the Korean Wiki Project has a lot of good sound files for various character combinations.

3. Billy Go's two Korean Made Simple books are just great. They're highly accessible introductions to core grammar and core vocab, with short, digestible chapters that gently build on each other (and some useful appendices, like another very complete overview over sound change rules). This is the sort of hand-holding you need to start to be able to find your way around in sentences, form your own, and develop a solid basis from which to attack more advanced areas.

4. Korean Grammar in Use is another excellent book series. It doesn't do hangul or vocab, instead focussing on numerous grammar forms. It's well-structured and benefits from great page layout, and is exactly what you need after Go's books to progress further. It's in English, but written by native authors and originates in the Korean university world. It's a bit harder to get ahold off; I got mine from Korean sellers on Amazon without problems though.

5. The Integrated Korean series has a deservedly good reputation as being suitable for self-study and being well-rounded and comprehensive. I agree it's good material that you want to look at, especially for practice (it has work books), but I don't think they're as accessible as Go's books, and not as well-structured as Korean Grammar in Use. I think it's best to start this series as supplementary fodder after the above.

6. I've used Anki flashcards for vocab training, but I don't feel like flashcards work well for me. I find it hard to retain words without sufficient context. The vocab I pick up while reading about grammar sticks much better. To really bulk up on vocab I recommend studying enough grammar so you can dissect even more complicated sentences, and then start reading prose with the aid of a dictionary. Get some children's book or YA fiction and work your way up.

7. Make sure you listen to some audio-only resources, not just TV shows. Some of the Korean phonemes are superficially similar to Western ones, but formed with slightly different tongue positions, and because of the McGurk effect (https://www.youtube.com/watch?v=G-lN8vWm3m0#t=86) you can actively hear them wrong when you look at speakers' faces. Once you know enough hangul and grammar and vocab to have a hope of keeping up with spoken Korean, bulk up on audio. TTMIK has a lot of conversational resources. That said, watching TV shows and stuff is of course good fun anyway, and especially variety shows often have on-screen subtitles featuring the key terms of what's being said, which makes for great learning material ...

8. Subscribe to the /r/korean sub-reddit and read the questions and answers/discussion. You'll learn a lot from that and make lots of interesting connections.

9. For practice, use Hello Talk (chatting with natives) and Lang 8 (write longer texts and get corrections by natives). There's of course also "go to Korea and immerse yourself", but not everyone has that luxury.