imo, a partial lease is probably best to start - one where you pay for being able to ride/use the horse on specific days of the week. That way you aren't throwing yourself straight into the costs of "ownership" as you might in other circumstances.
If you were to free lease (as in the example I gave earlier), if the horse was to have something happen (colic? Injury? Etc.) YOU would be responsible for any vet/farrier bills. In a partial lease that would be spelled out as to who is responsible and I would recommend you making sure that the owner is listed as responsible for such things UNLESS you were the cause of said injury.
Just my opinion.