I'm going to show you how to do the long tail cast on. For that you're going to need a long tailed yarn, so find a tail of yarn from your ball of yarn and sort of stretch it out. For this example we're not going to talk about really how long it needs to be, but you need a sufficient amount of yarn to get through the number of stitches you're going to cast on.
So for this cast on you do not need a slip knot on your needle. You simply need to take the long tail, put the needle under it and hold it with your index finger. So I am holding the needle in my right hand and I'm using right index finger to hold the yarn down. Then I insert my left hand pincher fingers in between the the strands of yarn, keeping the long tail towards my body and the ball end away from my body.
So my pincher fingers go in, I grab both pieces of yarn with my other fingers and I pull up and back, loading it like a slingshot. Then my right needle tip is going to do two things. It's going to grab the yarn that's closest to me on the outside of my thumb. It's going to grab the yarn that's over my index finger, and it's going to come straight through the loop, and you need to pull tight.
Again, reset your slingshot, grab the yarn that's on your thumb. Grab the yarn that's on your index finger, come straight through the loop, and pull down and reset. Now I actually never let go of the yarns that are being tensioned by my bottom two fingers. That's just because I've been doing this long enough that I make it fast and easy for myself by not letting go.
But if you're new at this it's totally okay to need to reset your hands every time. So that would look like this. Pinch your fingers in, grab the yarn, pull up and back really making this a nice, tense slingshot. Go over the yarn on your thumb, over the yarn on your index finger and then pull down and tighten. But the longer you do it, the better you'll get at being able to keep your hands basically in one place, and use that thumb to really open up your stitch.
Now don't over-tighten these stitches. They should be able to slide really loosely on your needle. Pinch your fingers in, open up, grab the yarn and pull back, around the outside of the thumb yarn, around from the top of the index finger, straight through that loop that you've just made and pull down to tighten. That is the long tail cast on, which does get easier with practice.