Today I figured I would start with a project that would take the longest. Knitting/crocheting a scarf or other small object. In the past I have made small coin purses for my friends, scarves, and a truly hideous blanket (that I never actually gave to anyone out of pure shame). For beginners I would recommend really simple patterns as well as felting. When you felt it shrinks what you have made and covers up any mistakes.
Photo courtesy of Answers for Insight
Image courtesy of feltcafe
Image courtesy of mi_entropia
Another great thing about knitting a gift for someone is that you can personalize it. And just remember, your mother and your grandmother will appreciate anything you made, no matter how completely ugly or awful it truly is.
Image courtesy of rubies and pearls
I typically find my patterns online or in books from the library. One of the most helpful books for me so far is Stitch 'N Bitch. I would also highly recommend stopping into any local knitting store (I do not mean big name craft stores, but rather local, specialty knitting shops). They will not only help you pick out the right yarn, but they can help you find a pattern and even provide support later on while you are knitting.
I really like Knitting Help when I am lost or need some advice. To be honest I have tried to learn how to crochet, but it's still mind boggling to me, so I do not have any advice on where to look for help.
Want to give something homemade but don't want to actually make it? Check out etsy for incredible hand crafted items.