This is a personal viewpoint for those going on the self teaching route.
It's not a short post. Go make a cup of tea before you start.
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I can't advise you on a cheap keyboard. What I can offer are some suggestions to help you choose one. And a book.
Get to a piano store and try the following on a good quality upright such as Yamaha U1/U3 or Kawai K3/K5 or that six figure concert grand if the salesman doesn't mind.
Play slow repeated notes to see how many volume levels you can discern between softest and loudest, the forces required between the two extremes, and the controllability - how many volume levels YOU can make repeatedly.
Listen to the different tones. Soft notes are close to one tone from the initial attack to the beginning of its fading away. Loud tones have a strong, biting attack then quickly settle to an even tone before beginning the fade out.
Listen to how long the notes sustain at the upper end of the keyboard, the middle and the lower end.
Learn a simple melody such as Beethoven's Ode to Joy (the European anthem) - just one finger will do, we're not trying to impress anyone here - and play it as many different ways as you can trying to simulate a sweet, gentle flute, a melancholy violin, the garlic tones of a bassoon and the strident tone a trumpet. Listen to how malleable the sound is on a good quality piano - despite your somewhat limited technique.
Then do the same on a good quality digital such as a Yamaha Clavinova, Roland HP series or Kawai CA series. You now have a baseline against which to measure your prospective budget instrument.
Try them out and see which one gives the least compromises for the least money. At the beginning be more concerned with the feel than with the sound. The sound is what it is - deal with it. All it has to do is let you know when you've pressed the key hard enough or soft enough. Bob Dylan doesn't have the voice of angel but he can invoke passion and express music when he interprets a song.
The action, the feel of the keyboard is what determines your relationship with the instrument and, if you're going to be learning on it for some time, all the other pianos you're ever going to play. Ever! This is why people recommend good instruments.
A Clavinova is a big outlay but if you keep it for ten years, and I've kept mine for 23, it's going to cost less money than the tuning costs of a regular acoustic over that time. In other words, after ten or so years it's free! It won't take you to Carnegie Hall but it will take you as far as most people are able to go in a normal lifetime.
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Now, as to books, I don't know many method books but the Faber and the Alfred series have a lot of devotees here and I'd take that as a strong recommendation. But I do find these method books slow.
I would recommend you tackle pieces like Burgmüllers Op. 100, Bach's Anna Magdalena Notebook and Clementi's sonatina's well within the first year.
You don't have to play them fast. You don't even have to get as far as half speed. You do have to get used to playing scales passages, broken chords, broken thirds, arpeggios and trills and interpreting them. I also believe you have to get used to memorising them. As a supplement to your method books do two bars a day of more advanced pieces (not advanced pieces, just more advanced than the method books) and start memorising them.
Play the same two bars, dead slow, every day until you have them in your head and your fingers then move on to the next two bars. Play over the bars you've memorised at the weekends and, trust me, the speed will come of its own accord. You never need to rush memorised pieces. You will eventually play them at the speed you hear them in your head. You don't ever have to play any faster than you can control. Once they're memorised they all have the same difficulty level - none. It's like having the answers to a quiz. You know the answers or you don't - they're not hard because you know them. They might have been hard to learn but that's over now. Once they're in your head you can keep them forever. And you can play them as easily as you recite your name and address.
This will give you scales and arpeggios in a musical context and by memorising them they will take deep root in your fingers. There is no better or faster way of improving your overall technique than memorising pieces replete with technical passages.
You also have to get your hands working independently for handling polyrhythm, like quavers in one hand and triplets in the other or a shuffle rhythm against straight quavers.
Set your metronome for 60 to 80 bpm. Slower than that and it's difficult to judge the timing. If you have an electronic one set the beep to sound on every fourth note. (If you have a mechanical one get either an electronic one or a drum machine.) Find the shortest note in the piece, usually a semiquaver (sixteenth note), and play them every fourth beat. If the metronome is going beep-dit-dit-dit play the semi's on the beep and use the dits to prepare your fingers. This might sound slow but it imparts deadly accuracy. As long as you always hear it and play it with this kind of accuracy you will seldom hit wrong notes. Once you have a long enough section in your head, around eight bars or so, let the tempo develop until it's about right for the piece as long as you maintain accuracy.
Pieces up to moderato I usually play up to tempo nearly all the time, faster pieces I play around half to three quarters speed nearly all the time. I hardly ever play fast pieces up to tempo except at the weekends. I will not accept wrong notes. I always finish with a slower tempo before I put the piece to bed.
If you don't have a teacher listen to as many good artists as you can, for free, on YouTube. Watch close ups of the hands for all the different ways the fingers hit the keys, and all the different ways the fingers leave the keys. Spot the difference (and hear it) between playing from the fingers, the hands, the wrists, the forearms, the shoulders and from the base of the spine.
I do recommend this book. It's not a method book, it's not a list of pieces. It's not cheap. It is good. It covers technique, phrasing, practising and performance. It won't replace a teacher but it will cover most of the gaps and help avoid pitfalls.
http://www.amazon.co.uk/Principles-Technique-Interpretation-Kendall-Taylor/dp/0853600732My copy is falling apart at the seams. I can't give a better recommendation than that. I had a teacher for eight years, a damn good one too, but this book has got me through some difficult patches that I've had since then and even during my lesson years.
). So, if you don't know anyone who can show you "how to play", you will need to supplement the book with a lot of youtube watching and it's not always easy for a beginner to discern between the good and the bad tutorials (I would personally recommend you this series of tutorials: 
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