I would certainly look into buying a car for a three month trip. In most cases it works out cheaper than a rental and you can choose the car you want. Check
www.cars.com for used car prices, but you'll need to check the Department of Motor Vehicles in each state to work out the cheapest place to register and insure a car. Generally used cars are quite cheap in the US although insurance costs can be more than you expect.
In my experience, US rental car companies only have automatics (although Adventures on Wheels have some manuals if you request this in advance) and have the habit of upgrading you to something the size of a bus (even their small cars are what European rental companies would call medium-large).
Auto driveaways are a great deal for getting across the country cheaply but time limitations may make this impractical for a longer trip.
The Northwest between Boston-NY-Philly-Baltimore-Washington DC-Richmond VA is well served by trains and buses and is probably easiest without a car but a car is essential everywhere else in the USA.