My partner has always struggled to learn knitting when being taught by right-handed people. After a time she developed her own style of left-handed knitting and I've encouraged her to share it with you all to help any left-handers out there.
Step 1: Casting On
Begin by tying a double-knot around one of your needles, giving yourself enough room to fit both needles through.
Take the needle with the loop in your right hand - this becomes the first stitch.
Slide the needle in your left hand through this loop from underneath. (Fig 1)
Take the wool around your left hand needle - go from the back, around the left side and then between the two needles above the stitch. (Fig 2)
Here comes the twist (you may want to keep a finger on this latest loop as you get used to knitting). Slide the left needle down and bring it through the stitch towards yourself (Fig 3) - keep the loop you have just made on the needle but pull the needle under and in front of the original stitch - see (Fig 4) for how it will look after this step.
To cast on slide your new stitch on the left hand needle onto the right needle. Repeat the above using the new stitch. Keep going until you have cast on the required number of stitches, this is your first row.
Step 2: Knitting a Stitch
Knitting is very similar to casting on in that the stitch is the same.
Begin with your first row on the right-hand needle.
Slide the left-hand needle up and under the first stitch, coming in from behind any 'clutter' you have made in your previous row - make sure you don't catch this in your new stitch.
Take the wool around your left hand needle - go from the back, around the left side and then between the two needles above the stitch.
And twist. Slide the left needle down and bring it through the stitch towards yourself - keep the loop you have just made on the needle but pull the needle under and in front of the original stitch.
The two needles are now joined by what looks like a figure-of-eight, now simply slide the right hand side off the needle - be careful you don't slide any other stitches off. You should now have transferred one stitch from your right needle to the left and you may notice this side now has the beginning of an additional row.
If you are following a pattern that tells you to inc (increase) or kfb (knit front&back), instead of sliding the stitch off at the figure-of-eight stage, re-use the stitch on the right needle so that you have 2 stitches on the left for every one on the right - this gets very difficult if you don't have enough slack in your piece.
Step 3: Purl
The typical pattern you see on knitted items is from alternating rows - knit one row then purl one row. If you are doing any increasing or decreasing this tends to happen in the knitted rows only.
To purl you will need to come in from above pointing down and in front of the stitch (also any previous rows).
To take your wool around, begin under your left needle, take it towards the left needle tip and around the back of the left needle only - if your right hand gets in the way the first few times you're probably doing it right.
This twist is slightly more difficult than the knitted twist - this time you're twisting the left needle backwards. Keep your new loop on the left needle but pull it out of the original stitch. I find it helps to slide the left needle tip up the side of the right needle to avoid this loop falling off. (Picture 2 to picture 3 sums up this move).
You can now go ahead and slide this stitch off your right-hand needle, keep working down to the end of the row.
Step 4: Casting Off
Casting off is a very satisfying process and very easy once you have the basic knitting stitch mastered.
Firstly, knit 2 stitches onto your left hand needle.
Now, use the tip of the right needle to carry the first stitch (lower down) up and over the second and off the needle - be careful to keep the second stitch on the needle.
Knit another stitch onto the left needle and again slide the lower stitch up and over. Keep going until you have one stitch left and it's on the left.
Now simply cut the wool and pass the end through your remaining stitch, pull the piece off the needle and pull the loop tight to close it (you may need to leave a long end for sewing depending what you are making).
Congratulations on making your first piece.