I have tried growing plants from their clippings and am not having any luck.
The following articles should be of some help to you: https://www.gardeningknowhow.com/garden-how-to/propagation/propgen/types-of-plant-propagation.htm
https://www.gardeningknowhow.com/ornamental/trees/tgen/how-to-root-cuttings-from-various-shrubs-bushes-and-trees.htm
What is soilless mix and where would I find it?
This article will help: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/soilless-growing-mediums.htm
I understand there are many types of rooting hormones. How do you know what type to use?
It depends on the plant you want to propagate or clone. There are specific rooting hormones for certain types of plants and which you can check out by checking the label.
How do you take a cutting and get it to grow successfully from most plants?
This usually depends on the type of plant you are getting the cutting from. These articles should help: https://www.gardeningknowhow.com/ornamental/trees/tgen/how-to-root-cuttings-from-various-shrubs-bushes-and-trees.htm
https://www.gardeningknowhow.com/garden-how-to/projects/rooting-plant-cuttings.htm
In your section on how to root hardwood and softwood cuttings, one of the directions says to place the branch in a small pot of damp soilless mix. What is soilless mix? Please give an example.
This article will help: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/soilless-growing-mediums.htm
Can you regrow a plant from fresh vegetables or fruit?
The best answer is it depends on the cultivar. Most vegetables can be grown from the plants seeds or cuttings. Fruits can be regrown from budding or
grafting onto new rootstocks. Do you have a particular item in mind?
How can I tell that my new tree cuttings are healthy and rooting?
In rooting cuttings, patience is the first rule - after keeping them moist. So don't let them dry out, and wait. When their time is right, little greenshoots will begin to appear. If it's been weeks and weeks, you can lift them very gently from the rooting medium, to see if roots are forming. If the cuttings are blackened, soft, withered, or the tops are crispy brown - no greenwood - I'm afraid you'll need to try again. If, of course, they still live, replace them with utmost care, and wait some more.