We did sing and sign. Going to the classes was really good in terms of making sure I knew the signs well, and it's much easier to learn them embedded in songs (and also DD loved them), but reinforcing at home was really important. Having a DVD also helped as DD is obsessed with the DVDs and she has picked up a lot of signs that I've never done with her just from watching them. I'm sure you could do it the way you describe, just by teaching yourself from a website, but it will be harder work than classes as the onus will all be on you to memorise the signs, decide which ones to use, and make sure you're using them often and consistently. Courses will usually let you do a free trial session before you commit, so worth going along once and seeing how you find it.
If you go ahead, bear in mind that there is potentially a long gap before the baby has the dexterity and understanding to sign for himself/herself (depending on how old they are when you start), and you need to keep going, and remember they are taking it in long before they can sign back. I did a term of classes when DD was 6 months. She suddenly started signing just after her first birthday.
As pp said, start with some signs that are simple and that you can use several times a day (eg milk, eat, drink, all gone). 'Where' (palms up and move hands back and forth) is easy to introduce in a game of peekaboo or hiding toys and finding them, and also easy for them to copy. If your baby is old enough to be pointing at things, it's a good opportunity to introduce a sign, as they are signalling to you what they are interested in.
Baby signing has been amazing for us - getting the insight into how their minds work is just brilliant, and it definitely reduces frustration (except when DD makes up her own random sign and then shouts at me because I don't know what it is!)