Basically you are doing part of the work for the tree service in return for the free wood. You take what they give you which means ANYTHING. They will fall the tree and cut it up into 5 or 6 ft. sections, then you need to cut it up and pack it out. Sometimes they will just limb the tree and not make any cuts.
You need to have the equipment to get the work done, this means a large saw to cut up those large diameter trees. The large trees are those that the tree service would be more than happy for you to cut up for them and pack out of that backyard for them! May need a handtruck or wheelbarrow to pack the wood out.
Then the tree service and their customer will like it if they see you wearing PPE (Personal protective equipment) like chaps, hardhat, face shield, and ear plugs. If working for a pro, look and act like a pro.
To the customer of the tree service, YOU are an employee of the tree service doing work for the tree service, so you need to treat the customers with care or you will give the tree service a bad name. This means ask before doing anything like driving your truck on their lawn. Ask if they have lawn sprinklers and be careful to not drive on these and damage them. If the customer asks you to do something and this is a reasonable request, do it. (Or call the tree service and ask what to do.)
The customer wants the tree cut down today and they want it gone within a day or two. When the tree service calls you, you need to go out there today or tomorrow and have that tree gone within a day or so. You can't wait a week to show up and you can't take a week to remove the wood. Of course if there are a lot of trees, it will take longer, but the customer wants to see you out there everyday until it is gone.
Then clean up your mess. If you get sawdust on their driveway or sidewalk, sweep it off before leaving. Pick up and take any remaining pieces of bark limbs, etc. so you leave their yard clean. (Tree service basically takes all the limbs.) Rake sawdust on the lawn so it levels out and is not in piles.
Basically if you hired a tree service to cut down a tree and remove it, what would you expect? Well treat the customer in that manner and leave their yard how you would expect it to be left.
Then when you are done, call the tree service and tell them you are done. With some jobs, they don't get paid until the work is complete (which means when you are done).