First call the USDA for the state you are moving to. They will tell you what you will need to bring in livestock. There are different requirements for each species. Some of the tests must be negative within a certain number of weeks of the move, which means that unless the bi-annual mobile vet visits within the time frame it won't help you. Some of the tests st be sent to a lab so it will ake several days or weeks to get the results back, meaning you will have to figure how much time before you leave you need to do the tests. If it is for a herd health check for someone that is leaving the state, the local vet might do it for you as a one-time thing.
You will also need to check on whether you need to comply on health requirements (certificates) for each state you plan to travel through. Also do the states require transport slips? You need to make sure that you have proveable ownership paperwork on the animals - registration papers or bills of sale with you as owner that conform to tatoos, ear tags, microchips, or animal identification, etc.
Always pack enough food and water for a longer trip than you anticipate. Bring gallons of water from your own home to gradually get them used to different water. All water is different and has a different odor depending on the minerals in it. Animals can get sick drinking strange water they are not used to. Also bring at least 2 weeks of hay and feed with you since you will want to gradually shift them from your hay and grain to the new stuff available in your location.
If you don't have a stock trailer, check with the various animal shipping companies for quotes on shipping your livestock. It might be cheaper than buying a trailer, etc. You will still need to provide the health docs though.