I have never driven an animal across more then 1 state line to the destination, I used to receive and then reship animals after quarantine, which would be similar to what you will need to do, but it was always through the airlines. If you want my thoughts here is what I would do if it were me and my animals.
First do you have any friends who could hold the animals in California or Florida? It is much easier to have the animals shipped by you and held in CA until you arrive then it would be to leave the animals in FL and then have someone else ship them after you have a place here in CA. If this is not a possibility you need to find a animal hotel in CA that deals in pets other then dogs and cats and make arrangements to have them take control of the animals on arrival. It will not be cheap and you will have to fly them here as freight which is also not cheap but that is your only option unless you drive them yourself. You will need to find a freight forwarder who deals in animals and contact them unless you are a known shipper with the airlines.
There are companies who ship animals for a living, Air club for pets (http://www.airclubforpets.com/index.html) is one company that you could contact. I have never dealt with them but you can do a search online for others. I don't know the cost, I always did it myself, I'm a known shipper on several airlines and have used FedEx and freight trains as well, but there is no reason not to call around and get prices.
Your least expensive way to accomplish your goal is to drive them here, find out exactly the route you will take and check all of the states rules and regulations that are along the route, also check any state that you might have to go through in an emergency just in case. Follow each states guidelines and have all documentation necessary for the trip. Many states will need vet certifications, but all of them will probably accept the same vets results so just bring along copies in case you have to leave a set with the AG depts at any one check point.
As far as the animals on the trip itself, you will need individual cages for each animal, big enough for the animal to stand up in and turn around but not much bigger, large areas can cause the animals to try to move around too much and they can get hurt on rough roads or curves or whatever. Stop frequently to give the dogs and pig a bathroom and water break but the chickens can stay in the cages for several days with fresh bedding, food and water changes at least once a day and twice, morning and evening, would be better.
Unless you have someone else do it for you this is the best advice I can think of quickly, if I think of anything else I will let you know but I wish you luck, I do not ever look forward to shipping animals because once they are dropped off (if someone else is handling the move for you) you don't have any control over their care or well being so it is very stressful until the move is complete and you are reunited.
If you can think of any specific questions I will do my best to answer them.
