Propagation
Cuttings
This is a good method for things like ornamentals or woody shrubs. I have also had success with grocery store basil.
Rooting hormone is optional but may be necessary depending on your plant. The container you propagate your cuttings in should be at least 2 inches deep to allow for enough media to hold your cutting upright and let it grow its roots. You can propagate cuttings in various different medias, such as:
Water
Using water for propagation is good if you want to watch the roots develop, but it can cause shock when the plant is transferred to dirt because the roots it grew were adapted to water, not dirt, and the plant will show signs of dying, if not die completely- so a lot of attentive care is necessary. You have to regularly change the water to keep the oxygen supply high. A way of making sure less shock happens is propagating plants in a fish tank! Plants that like humidity will love this. The nutrients that the fish leave in the water are good for the plants, and the fish shouldn't nibble on the roots (too much).
Potting soil
Potting soil is what most people go to, but it tends to hold too much water and not enough air so your plant base or roots could potentially rot.
Perlite
Perlite has good water drainage and great air circulation, and it provides support so the cuttings don't fall over. It is also very light and almost fluffy, allowing roots to easily grow.
Sand
Not plain old beach sand- opt for something from Home Depot or something. Like perlite, it also has good drainage and airflow. Coarse sand is recommended, the coarser the better- but if your sand is extremely coarse (like, nearing gravel territory) you should mix in a small bit of peat to hold water better.
Peat and coarse sphagnum moss
Peat moss is acidic and holds moisture very well- sometimes too well, as it can stay too wet. Coarse sphagnum moss also has good moisture holding abilities and it has good aeration when you don't compact it, though it can be difficult to re-wet if it completely dries out. People like to mix peat moss with sand or perlite.
Some misc tips
Retaining moisture is IMPORTANT. Cuttings have no roots, meaning they have a very limited ability to take in water.
To reduce water loss, you should start with fully hydrated cuttings- water the parent plant the day before so they're hydrated and ready- and get your cuttings into the prepared containers as quickly as possible.
You may also find it helpful to trim down large leaves or even remove some leaves from your cuttings altogether, to minimize the area from which the cutting will lose moisture.
Keep the plants in an environment with high humidity. This can be done by putting a clear plastic bag over the plant, leaving it partially unsealed so that air can be exchanged. You can use wires, chopsticks, or other methods to prevent the plastic from touching the plant. You can also use a clear plastic bottle, sliced in half and placed over the plant like a dome, with the bottle cap left off to allow air flow.
Be warned, though: as cuttings need a lot of sunlight to grow properly (6-8 hours of bright indirect light is recommended), the temperature inside the domes/bags can get too hot. Make sure they maintain an air temperature of 65-75 degrees Fahrenheit, or 18-23 degrees Celsius.
Rooting hormone can help your plants develop faster, but too much rooting hormone can slow root growth! Rooting hormone comes in powder and liquid form. Make sure to read the directions on the label! Rooting hormone keeps for a very long time but you should replace it after two years as the effectiveness drops after this point.
How to propagate via stem cutting
This example uses grocery store basil, but I'm 90% sure it's applicable to other plants as well.
On your chosen plant, each bundle of leaves comes out from a little knob in the stem, this knob is called a leaf node. Choose a stem (or don't if your plant just has one stem) and count the nodes, start at the tip and move towards the base. If there are more than three nodes, cut the stem just above the third one. That bit you just cut is your cutting! Keep doing this process until you've taken as many cuttings as you want/as many as the plant has to offer.
Quick Tip
Always leave at least one leaf node on each stalk, because the plant can't make leaves with no nodes, and without leaves it will die. You want the parent plant to grow stronger. If a stalk does not have more than two nodes, don't cut it, just leave it to grow.
Remove all the leaves on the cutting EXCEPT the ones on the top leaf node. Also remove any flowers or fruits it may have grown. If you want, you can also treat the cut end with rooting hormone.
Stick it into your media of choice. Make sure all the leaves and at least one node is free from the media.
Water the media (unless it's water) then put it in a warm spot with bright indirect sunlight, such as a windowsill. You want it humid too so this is a good time to put your plastic bag or bottle over top.
Leave them to grow for 3-6 weeks. Check them frequently- water them if they're dry, and if they look sad adjust the conditions they're in. If they are growing tall and thin, but the plant itself is meant to have many stems, you can encourage these side stems to grow by removing the small top leaves.
When roots 1 inch or longer have formed, replant it in potting soil. Keep it well-watered and in bright indirect sunlight, but no need for the humidity dome anymore.
Within a few weeks it should be okay, and you can move it off the windowsill into its preferred temperatures. You can also begin to fertilize it at this time.
misc
How to propagate onion and garlic
You need:
- Onion or garlic
- Sunny spot
Peel your garlic, or grab your whole unpeeled onion. Put it in a sunny spot. Leave it and wait.
Eventually, maybe after a week or two, it will sprout. Then you can plant it.
Quick Tip
Garlic doesn't really have to be in a sunny spot, you can just peel it and leave it alone for a while. I've seen garlic sprout in a sealed ziplock bag in the refridgerator. It doesn't care.
Here's a great post about transplanting trees!