Hydrangeas are one of the easiest things to propagate. They're so ready to grow that it is pretty hard to mess it up (though it certainly does/can happen!)
First things first:
There are several types of hydrangea. I have Hydrangea paniculata which doesn't mind (in fact, it thrives) in the full sun of my backyard. The other hydrangeas are part-shade lovers, and though they're gorgeous, I just don't have anywhere to grow them.
Also, Hydrangea paniculata flowers on NEW wood, wood that grew this year. The other hydrangeas flower on OLD wood. Since it flowers on new wood, I can take cuttings in the spring and not lose my flowers for the year. In fact, pruning now will give me a much healthier plant that I can manipulate it's shape. It can even be grown in a tree-form, which the other hydrangeas do not do.
And lastly, it is illegal to propagate any plant that is under patent. Before you propagate anything, do a google search for the name of your plant and add "patent" to the end of your search string. Review the results you get to determine if the patent is active. NOTE: patents expire after 20 years. If the patent was obtained in 1973, for example, it expired in 1993 and you're free to propagate. NOTE #2: It is even illegal to propagate plants under patent for your own use. It doesn't even have to be a plant for re-sale.
Ok, so here we go.
First, cuttings can be taken about six weeks after the plant starts putting out leaves. These will be softwood or semi-hardwood cuttings. This is a great time since the plants haven't set flower buds yet. Fall/winter is a good time as well. Same technique, but using hardwood cuttings. Rooting takes longer - maybe even waiting until spring - but they still work.
Here's an example of what I'm taking cuttings from in my garden. This little guy was planted last year, and was much smaller. Paniculatas grow fast and large. Don't worry about taking cuttings from a small one this size, or a huge one. They're hardy plants and can take a beating. Just be sure your scissors/pruners are clean and sharp. You don't want to crush the cut location.
Next, here's the Chantilly Lace hydrangea after I've finished taking cuttings and the tray of cuttings next to it. Hopefully you can see that I went quite a ways into it. Each cutting was about 1 - 1.5' long. Where before it was a mound like the above picture, it now looks rather flat. Don't worry - each place that I took a cut will grow two more branches, resulting in a fuller plant and more flowers. It just looks a little funny right now.
Keep in mind that I'm taking 100+ cuttings from each plant. If you only want to grow a few for yourself, you'll probably only need to take one or two branches and your plant will probably not show it at all.
After you've taken your long cutting, put it in cool water immediately and bring it to whatever you're using as a rooting container.
Fill your container with *GOOD* lightweight potting soil but don't press it in too firmly. You want it to be very light. I use Pro-Mix Organic Vegetable & Herb for just about everything. It is pricey, but the quality is obvious in the result. Trust me, I've tried about everything.
As for your container, I'm using 50-count deep plug trays. You can use anything from a yogurt container to a regular old plastic pot. Just be sure it has drainage holes.
Now, take your long cutting and identify where the nodes are (that's where the leaves come out). Cut your long cutting into pieces that have 2-3 nodes, making sure you know which end is up.
Remove the leaves from the bottom nodes so that there are only leaves on the top node. In this picture, (hopefully) you can see the leaves at the top, a node just under my thumb, and another one at the bottom of the cutting.
Now cut off 1/2 - 2/3 of the each leaf. Because the cutting has no roots, you want to slow down transpiration and the plant's need for water. Removing most of the leaves accomplishes this.
Stick your now prepared cutting into your soil-filled container so that just the top node (with leaves) is above the soil.
Finally, water well! I actually place the tray/container in a shorter tray and fill that lower tray with water in addition to watering the top. This ensures that the soil medium is moistened all the way through without having the cutting sitting in moist/muddy soil (which will cause rot). After a couple of days I take the tray out of the sitting water so that it can drain. We don't want to drown or rot our cuttings!
Now remember, your cuttings have no roots yet and they've been through quite a bit of trauma. Place your tray/pot in a semi-shaded location...under a tree, under other plants, on the porch...and make sure the soil stays moist. NOT WET. It should *not* be muddy. Never muddy. My sprinklers are set at 10 minutes per day, twice per day in hot, sunny weather. Do not let the soil dry out. These cuttings need humidity! If you are in an arid location, put a plastic bag or clear-ish plastic container over your cuttings and vent it daily to keep the humidity up. I find that I don't need to do that here in Georgia.
After about six weeks your little cuttings should be pretty well rooted and showing new growth at the leaf axils. I'll post a follow up later when it is time to transplant the babies.
I hope this helps! If you have questions or I need to clarify, please let me know in the comments below and I'll do my best to help you out.