Most plants can be propagated. Many plants (perennials) can be split. Others can be rooted by taking a cutting (you want to cut the tip and not the woody part).
With perennials it's best to divide them in the early spring or early fall. Dig a large area around the plant to get most of the roots. I then fill a large bowl with water and let them soak for a while before I divide and transplant them. You can put the separated plants into pots until you decide where to transplant them.
Tips for transplanting here.
For stem cutting, snip at least 4" from the plant, trim the lower leaves and place the plant in water until you see roots. For added help, dip in hormone powder before planting in moist soil.
Get together with neighbors and swap plants to give you a bigger variety in your garden.
Plants I have had most success with here in Texas are:
Rose of Sharon (these sprout around the tree and you can pull them up, put them in water for a day or two then plant them.)
Mexican Petunia - These spread and grow up to 3 feet so if you want to contain them plant them in a pot. You can simply pull these up, soak in water and replant.
Dwarf Mexican Petunia - These grow in clumps and can be easily dug up and split. They're great for borders.
Sweet Potato Vine - These are annuals but I have been able to keep cuttings through the winter (in a vase of water). You don't have to buy loads of plants if you're patient. Start one and take cuttings, place in water until you see roots and then plant. They do need to be watered regularly once in the ground.