You have recieved some interesting and often conflicting advice (and you will continue to get this kind of advice for many years). I am a gifted/talented teacher with a 4 year old son in a very similar situation. There is the option of early entrance to kindergarten, but I very rarely suggest it. Kindergarten will be a whole new world of friends and routines that your son will enjoy. The academic portion will be much too easy, but it will bring the understanding of how to work in school and give him experiences with a lot of fine motor control (this is often lacking in gifted kids, but not always - you didn't mention this). As for a gifted pull-out program there will probably not be any option for a few years, if ever, in your school. Continue to be his advocate for challenging education while balancing a nice-mommy approach. Unfortunately in many schools the parents who push the most are pushed away by administrators, overwhelmed teachers and other parents who will think you are bragging (you're not). One trusted approach is to ask his teacher what you can be doing with him at home - most of his quality education will come from the experiences you provide for him - not the direct teaching or drill and practice, but the places you visit and the conversations you have with him).
Grade skipping, when done well can be very effective, but there are a lot of challenges in the school system. It is a proven method of teaching children at the level where they are most ready, but the teachers, school support system and administrators must be in the on the conversation and your child should have a self-esteem that is already strong - it can be a seamless option, or it can lead to disaster. Tread carefully - if this is right for your son, it will be great, but if it is just to satisfy a desire to further a "my-son-is-brilliant" need (which I don't think you have), then no one will win.
Talk to him a lot, read everything, answer all his questions (often with "what do you think?" or "how can we find out?") and use the biggest range of vocabulary you can - recent Harvard research says that the vocabulary that a child has at age 5 will directly relate to the size of his vocabulary in his 20's!! Stick to shows on PBS - Sesame Street, Super Why, Sid the Science Kid, Between the Lions, Word Girl, and Fetch. Keep him interested in science, history and mathematical reasoning and enjoy the things he can do.
Finally, be careful who you are listening to - you may choose to take the advice of those who say what you really wanted to hear, but it may not be the best advice. Loading your son with extracurricular activities and classes may not be what YOUR child needs or wants.
Good luck and feel free to write back - my aim is to be very open to parents and their ideas about their own children - even if they're not in my district!