How to grow cuttings depends on the species. Some grow better taking cuttings from soft wood, some from hard, some can even be grown from pieces of root, though it's not something I've tried yet. Here is a link to some information on cuttings:
http://http://grow.ars-informatica....hp?subject=cut0
If you search you can find lots more on this site and elsewhere.
Here's another idea if you're really patient - some of the fruit you eat may have suitable seeds to grow bonsai - I've planted a few citrus pips and a couple of litchis. It's not the right time of year for that though, but it's something to think about next summer.