Since horses are considered livestock by the Department of Agriculture (and we horse lovers totally disagree with that!)
Sorry it's off topic, but I found your comment curious. Having horses classified as livestock provides many advantages (especially tax advantages in some states). My state recently floated a bill which would have had the effect of declassifying horses as livestock... all the local horse organizations fought it hard.
To be on topic... in addition to the already provided good advice, you should also check your own state DMV. You will probably need a DOT number and commercial tags for your vehicle since your are hauling for hire.
Check with your insurance agent as they would best know the requirements in your state. They can also advise on the most appropriate coverage for your situation.
For the trailer tags, you may want to look at Maine. You can register (and title) your trailer in Maine and likely save some money.
You should also check out your state's rules for paperwork. You will typically need a negative coggins report when traveling in-state. When crossing state lines you will usually need a 30-day health certificate as well. These rules can vary, so you need to know what you will need for your own state as well as the destination and any states you are passing through. The owner of the horse would provide that to you, but you are still the one who is responsible for having it when you are hauling.
Also be aware of what paperwork your destination requires. Let's say the require a rabies certificate and the owner didn't provide one... you could be stuck with a horse you cannot deliver.
If you are out west you also need to learn about Brand Inspection requirements for various states. Basically if you are traveling from, to or through a Brand state, you will need additional paperwork. If you are in a Brand state, check your local office for details and requirements. If you plan to drive through to to one, check with that state's office. Again, this is something the owner would provide, but you will be the one in trouble if you are hauling without the required paperwork.
Hope this helps.