Have you tried asking for tips at a makeup stand in a large store? They usually do makeovers ( I'm not sure if you have to pay for them but it's worth a go ) Try a MAC stand, their makeup is expensive, but it is really good quality stuff. Their eyepaints are wonderful, and the pigment pots last AGES as you only need a small amount to do a whole eye. I'm sure the people who work on these counters can help you with application techniques, and reccommend what products and brushes /applicators to use.
When it comes to eyeliner, start off with pencil liner first, to train your hand to stay steady. Resting your elbow on something, even if only your other hand is helpful. Once you have mastered pencil, try doing your design ect in
light pencil, then go over the faint lines with liquid liner for a heavier look ( like tracing pretty much ).
I always found the liquid liners with a sponge pointed applicator easier to work with than the really thin long brush you sometimes get, but that's just me.
I personally use a type of liner ( Babecake by Benefit ) that looks and behaves like a watercolour paint, I use a very fine short length paintbrush, wet it, swirl over the liner cake to coat it and then apply. I also use it for my eyebrows
I have green eyes too, and have found that deep deep red, purples, plums and surprisingly, mettalic emerald green, work really well for me in making the green of my eyes stand out. I also use a lot of black of course !
Hope some of that rambling is useful