Considering the traffic, the confusing roads and the ridiculous cost of parking, we don't recommend using a car to get around Tokyo. Day trips can easily be done by public transport, though having a car can expand your options.
You will need an International Driving Permit, which must be arranged in your own country before you leave. Play it safe by getting a copy of Rules of the Road (digital/print ¥864/1404) published by the Japan Automobile Federation (www.jaf.or.jp).
Rental companies with branches around the city include Nippon Rent-a-Car (www.nipponrentacar.co.jp/english) and Toyota Rent-a-Car (https://rent.toyota.co.jp/eng/). Expect to pay ¥8000 per day for a smallish rental car.