I have done signing with both my children. Both of them are way ahead of their non-signing peers with their vocabulary (and eldest is 8).
Baby signing has many benefits. Babies can communicate much earlier because they can make hand shapes much more easily than forming word sounds. As a result, they have fewer tantrums etc which are borne out of a frustration at not being understood (plenty later on when they know you understand them and are saying no though!). When they are tiny, it starts them understanding that these weird sounds you make actually have a meaning, and it instils the fundamental basis to understanding that people communicate, and that they will be able to communicate too.
Signing actually helps their speech development, as it causes you to slow down when you are speaking to them (as you sign simultaneously), therefore enunciating much more clearly and showing the shape your mouth makes in forming those words. It emphasises the key points of the sentence, helping them to understand what you are saying. When my daughter was little, the way she said "milk" and the way she said "more" sounded the same. Because she signed as well, I understood what she was saying, but I was also able to repeat the word back to her to help her learn how to say it correctly.
I used to spend a lot of time playing with them using soft blocks etc with different animals on. I'd point to different animals and sign them. As they got older we had a game where I'd ask them to find certain animals and they would. They would then ask me where the pig was so that I could find it too. They learnt interactive play that way.
Nothing came a s a surprise to them. If we were playing and then it was going to be bath time, I would tell them that we were going to tidy up and then have a bath. They learn about timelines (in a very rudimentary way obviously) this way, can anticipate when events are going to happen. It must be difficult for a small baby to understand why one minute it is playing and the next it is being stripped off and put in some water with no warning! Signs for nappy changing etc were very good when it came to potty training - they were both able to tell me they needed the nappy change (if I hadn't worked it out for myself!), or able to tell me they needed to use the potty later on. They were fairly verbal by this stage anyway, but it helped them make the connection between the feeling of a dirty nappy and what had caused it much earlier.
They are both very good readers, and I attribute some of this to early signing. As we were reading books they would point to pictures and sign them, or ask what the sign was if they didn't know it.
By the time my daughter was 2, she was able to name many many animals, as well as different types within the group (eg she knew if the bird was a chicken or a penguin etc). She could differentiate between different vehicles (car, plane, motorbike, police car, ambulance, fire engine, lorry, truck, digger etc etc...). She knew all the colours of the rainbow, lots and lots of different foods (including what she did/didn't like!).
It really does have many many benefits. One story I tell to lots of people is when my daughter was about 18 months, maybe younger, she had recently moved into a new room at nursery, and was with lots of children she hadn't been with before. She had an unusual way of signing "more", which was very distinctive. On the way up to the room, there was a whiteboard on the wall which listed the meals they had that day, and how much each child had eaten. She often had x2 next to her name for lunch. Over a couple of weeks, there were gradually more and more children who had x2 next to their names too. And parents asking nursery why there child was doing this strange movement with their hands - it was my daughter's way of saying "more"!
Finally My brother started baby signing with his first child (younger than my first one), but thought he wasn't interested in it and so gave up after a term. I was on holiday with them about 18 moths after this, and my nephew was sitting on the beach, quite deliberately hitting himself on the side of the head. He kept doing it over and over, looking at my brother. My brother told him to stop doing it, but he wouldn't. I told him it was the sign for hat. My brother got the hat, put it on him, and he went off and played happily.
It works. It is very good. If you have the opportunity to do signing with young children then I can't recommend it enough.
This is very long already, and my first post on here (been a lurker for a very long time!0 so I will stop there. But I could go on much more!