Yes, you can sprout year round, 365 days a year. Throughout the year I will sprout in fits and spurts, sprouting for a while, and then... not sprouting for a while. But February brings a flurry of sprouting and I never get tired of watching the dormant seeds sprout into little live plants.
My first sprouting jar was an old, gigantic olive jar from a restaurant with a cheesecloth lid. Next, I graduated to a "real" sprouting jar with a fine mesh lid. These days, my favorite sprouter is, by far, EasySprout. Easy to use, easy to clean, perfect results, every time. You can buy one from many places online, including Amazon and, my favorite sprout vendor, sproutpeople.com.
What to sprout? The question should be: what NOT to sprout! There are the obvious standbys, like mung beans and alfalfa and radish and broccoli sprouts. Then, there are bean and lentil sprouts. You can also sprout nuts! Again, sprouting seeds are available all over the place. If you want some unusual blends or ideas, visit sproutpeople.com! (I am not affiliated with them in any way; they are just the best place I have found for buying and growing sprouts, recipes, information, etc.) Sprouting nuts is supposed to make them easier to digest and it softens them a bit if you want to grind them up for nut milk. I have made hummus with sprouted chickpeas and it was surprisingly good, though it tasted quite different from cooked chickpea hummus.
What else can you do with sprouts? Well, the obvious things include using them in salads, sprinkling on soups, in sandwiches and wraps, and rolling up them up in your sushi! Sprouted lentils make a wonderful salad. Just mix up a bowl with sprouted lentils, some chopped red onion, diced tomato, and diced cucumber. Then, add the dressing of your choice. This salad tastes great with a vinaigrette style dressing, and also a tofu or creamy-based dressing. You can eat the salad plain, or on greens, or on a sandwich, or in a pita. Really delicious!
I just sprouted some radish seeds. These are sort of peppery sprouts, much like radishes themselves. Some pithy directions: Place your sprouts into a jar or bowl or your sprouter. The amount of seeds varies with the type of sprouts and the size of your container. With the EasySprout I use 2 Tbs of radish seeds and maybe 1/4- 1/2 C of beans. Cover with water and soak for 8-12 hours. Drain well. Then, rinse well with a steady stream of water 4 or 5 times a day. After rinsing, you must DRAIN WELL. Make sure the sprouts aren't sitting in water or they will rot.
Here they are on day 2:
Here they are on day 3:
Here they are at the end of day 4, when I have decided they are "done." For radish sprouts, I like to de-hull them a bit, which sounds like a big deal, but it's not. I just put the sprouts in a bowl and fill with water. 
The seed casings either sink to the bottom of the bowl, or they float. I just scoop out the hulls on top and discard, then scoop out the sprouts and lay them on a towel to dry out. You don't have to de-hull sprouts at all, ever, but I find the hulls of the radish seeds are a bit bitter and tough. I don't try to get every single one, just a general clean up. That's it; they're done! I store them in a glass or plastic container in the fridge. If they seem really wet, I might wrap them in a paper towel. I think I might put some on my pizza tonight!!
The seed casings either sink to the bottom of the bowl, or they float. I just scoop out the hulls on top and discard, then scoop out the sprouts and lay them on a towel to dry out. You don't have to de-hull sprouts at all, ever, but I find the hulls of the radish seeds are a bit bitter and tough. I don't try to get every single one, just a general clean up. That's it; they're done! I store them in a glass or plastic container in the fridge. If they seem really wet, I might wrap them in a paper towel. I think I might put some on my pizza tonight!! 
