Putting - Starting from Scratch

Lessons
Definitely get a lesson! Scratch that... Don't get a lesson, get several! Most people have just one then never follow it up yet have multiple iron or wedge lessons. Almost (well nearly) half the shots hit in a round are with the putter!

Sort out the eye position, stance, alignment etc etc etc

Fundamentals
No matter how personal something can be, the basics MUST be right. Without fundamentals you cannot personalise it. GASP can even be used for putting! A short pre-putt routine with a quick read and GASP works wonders!

Remember:
Grip
Alignment
Stance
Posture.

Technique
Try to go straight back and straight through (no matter what putter type you have you should not arc through the ball). You may arc a little on the end of the backstroke (I certainly do) but the idea of the putter head shape is to correct you to straight back and through, not to exaggerate the problem!

NO BODY MOVEMENT, the unit of the hands, arms and shoulders move as one. The body is a perfect solid base.

Pace Control
Accelerating through the ball is essential and pace is controlled by how far you take the club back.

Remember, however far you take the clubhead back, you have to go past the ball twice as much.

Try using this method to hole a 2 foot putt with a 2 foot backswing (4 foot through), then try one around 1 foot (2 foot through), it will probably still be too much! Use this to learn how to accelerate through the ball. I putt this way on the course!

Compare it to a 2 foot backswing and a 1 foot follow through to really drill in the difference between the two!

Drills
I suggest Jasons suggested drill too.

I used to do a slightly different one from about 4-7 feet, my record is 52 (IIRC) in a row from 4 feet. Those drill types of practice sort out your game!

My last drill is at home on the carpet, putt from 5-6 feet to a 5p coin. Can you beat 8 in a row making a little clink? You have to roll right over the coin not hit the edge and stop a foot or two behind at worst.

You have to practice GASP at home too, 5 mins a night is enough to make a change. Then do the drill for 10 mins or so, I used to drive my wife mad!

Training Aids
Cant comment on a mirror as not used one. However I recommend a putting rail, worth the investment.

V-Easy was essential while learning the shot. I mean CRITICAL! Must keep that V shape and NOT flick with wrists, the unit from shoulders to club head move together.

Long Putting
Worth putting the time in, I like to pop my V-Easy down about 2-5 feet behind the hole (depending on putt length). I then have to get the ball between the sticks and the hole (1 point), hole the putt and its 3 points. Anything long (touches the stick or hits it), outside the box (2 foot either side) or short is -1 point. Anything level with the hole and within 2 feet either side is 0 points.

Set a score then try and beat it! Try from 10, 15, 20 feet and beyond. Its a cracking drill that is a lot of fun!

I also played a game of "18" hit 9 putts from various lengths to various holes (we only had 5 so 4 holes got 2 balls or 3 holes got 3 balls). Need to score 18 or better to "win".

Chipping
As for those who mention chipping, you are spot on, raise your chipping game at the same time as putting by trying to get up and down from off the practice green (if allowed), I have been lucky to have many clubs allow this so Ive had great chance to do that. 9 balls, 18 shots is the target!

Get a couple of chipping lessons, try getting a series of 6 lessons, 2 putting, 2 chipping and 2 on pitch shots. I doubt you will be shooting the same score after that!

A.O.B
Last tip- Don't neglect bunker play, wedge play, irons, hybrids, woods, driver etc. Try and keep those at the same standard while improving your putting by giving them about 20% of your practice time. A couple of range buckets to keep your eye in plus some course time. I found my alignment with my long game drifted so I had to keep that in check while bringing up the short game.

Summary
That's pretty much a dump of my putting knowledge right there, it got me down to 11 without really being able to hit long clubs and using my 3 goals method. My short game pretty made up 80% of my scoring ability.