Kuwait has a very cheap and extensive system of both local and intercity buses. You can also use local taxis to get around, though these have no meters, so get a firm price before starting out.
Renting a car in Kuwait will cost about US$20 a day. If you hold a driving licence and residence permit from another Gulf country, you can drive in Kuwait without any further paperwork. Otherwise you can drive on an International Driving Permit or a local licence from any western country, but you'll also be required to purchase insurance for your licence, which will cost about US$35.