Kuwait International Airport is 16km (10mi) south of Kuwait City. Kuwait is not a particularly cheap place to fly to or from. The airlines and travel agents tightly control prices, and few discounted fares are available. There's a small airport departure tax that may not be included in your ticket. From the airport, taxis charge a reasonably steep flat fee to the city; buses make the trip a lot cheaper.
Buses operate between Kuwait and Cairo via Aqaba in Jordan and Nuweiba in Egypt. There are also international bus services to Dammam in Saudi Arabia.
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.
If you are renting a car and 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.