It's great that you want to learn some sign language to communicate with your friend's girlfriend. It's very important for members of the deaf community to feel included, especially when they interact with the hearing community.
Starting with the basics, such as the alphabet, is certainly the best option. You can memorize the alphabet in a matter of hours if you just sit and work at it. I would recommend an ASL (American Sign Language) dictionary, because it will contain the entire alphabet, as well as numbers. Then, you can begin working on words/phrases that you would like to use.
The Internet may be helpful, especially if you want to see sign language in use. Sometimes it is difficult to understand how to move your hands/fingers when reading from a book. Online you'll be able to observe and model the movements.
Once you have these simple tools, the most effective way to become fluent in sign language is to use it! When I first started learning, I would try to sign along with songs on the radio and when I didn't know a word, I'd look it up. You can also do this with your favorite movies. Also, spending time with others who use sign language is the fastest way to learn. I'd recommend spending time with your friend's girlfriend and trying to learn. She will probably be more than happy to help you.
Learning sign language is similar to learning any other foreign language. It has it's own grammar, syntax, and dialogue. There are also several variations, depending on who you're communicating with. Additionally, sign language involves many nonverbal communication methods and even lip reading. There is no quick way to learn, and the more effort you put in, the faster your results.
That being said, it's a beautiful, unique, and fun language and I wish you the best.