Renting, you do not want a big flight dchool in the US, you will not get take the airplane away for weeks on end.
Most airports will have smaller schools who also use airplanes from private owners, you would likely get one of those at a lower cost maybe a two or three hour daily minimum.
Make sure the insurance us good, buy renter insurance - remember to factor in the cost of hotel rooms if doing long distance flights or buy a tent and select airports carefully.
Hour building through renting is going to be very expensive - consider buying something cheap and cheerful and selling at the end, pikely to save you a lot.