from Hamburg Airport
Most likely you will fly to Hamburg, which is about 100km south of Kiel. From Hamburg airport (HAM), do NOT take the train to Kiel, but take the Kielius (owned by Deutsche Bahn, your rail-and-fly ticket is valid on the bus, however a train ticket from Hamburg to Kiel is not!) for a faster transfer (about 90 minutes) to Kiel central station. The bus stop is just outside the terminals between terminal 1 and 2 (bus stop B). Look out for the yellow/green or white Kielius bus. Most busses leave at 15 minutes past the hour. You can buy return or combi tickets from the bus driver to save a few EUR (20.50EUR one way, 36EUR return, 27.30EUR combi, 49.60EUR return combi ticket). A combi ticket "city" includes the taxi fare from Kiel central station to your destination in Kiel (WAK guesthouse, hotel, University). If you share the taxi to the WAK, it should be cheaper to not get a combi ticket and get a regular taxi at the train station (you can ask the bus driver to book one for you, but there should be taxis waiting at the train station). Note that you can not pay by credit card on Kielius, there are ATMs and currency exchange in the airport terminal. You can also take a city bus from Kiel HBf (central station) to the WAK (single ticket 2.60EUR). On Sundays busses in Kiel leave only every 30 minutes (20 and 50 minutes past the hour, bus 501/502 direction Strande or bus 900/901 direction Krusendorf) and a trip takes 16 minutes to Schleusenstrasse. Alternatively Bus 11 leaves at 5 and 35 minutes past the hour (direction Wik, stop Knorrstrasse or Auberg) which includes a bit of walking.
For 30.90EUR you can book a seat in a KielExx shuttlebus which takes you (and up to 7 other passengers) directly to your destination in Kiel (travel time about 60 minutes). You are supposed to book a seat about one hour in advance, but if the driver has empty seats, he will be happy to sell you a ticket on the spot, KielExx stops at the Airport at the same place as the Kielius bus (bus stop B). If you already have a Kielius ticket, the cost of this ticket can be subtracted from your KielExx ticket if you hand it in. You should be able to pay by credit card, but cash and EC work as well.
"luxurious" (and economic for 3 or more persons)
If you want a car for yourself (or several people arrive at the same time and share the car), you can book a limousine for about 80EUR. If three of four people share this car, it will be cheaper than KielExx or even Kielius. The limousine will not be available on the spot, you have to book it in advance. You need to provide your flight and a contact number for booking. The driver will wait for you at the meeting point in the airport and take you directly to your destination. If the passengers need to be dropped off at different destinations, every extra stop (up to 3km) will cost you 3EUR extra.
For your return trip, remember that there is major construction going on on the A7, so you might want to take an earlier bus to the airport.
from Hamburg city
If you stay overnight in Hamburg or arrive there by train, you should travel to Kiel by train as the trip will be faster and more convenient than going back to the airport to take the bus. Kiel is connected to the German high-speed train network (ICE), the trains stop at Hamburg central station, Hamburg Dammtor, Neumünster and Kiel and the travel time is about 75 minutes. A ticket costs 29EUR one way, but you can book a "Sparpreis" ticket in advance which is valid for a train at a certain time and the ticket will cost you 19EUR, but don't miss your train! The IC train costs 23EUR but you can also get a "Sparpreis" at 19EUR. The IC is supposed to be slower than the ICE, but it seems to be actually 8 minutes faster (there is construction on the rail network in Hamburg. If you arrive more than one hour after your scheduled arrival time, you are entitled to a rebate, ask at the information counter for details). A standard ticket for the regional trains (RE, RB, but also private trains: NBE, NOB) costs 22.80EUR (and should allow you to use the bus in Kiel at no extra fare, ask your bus driver) and takes between 75 and 90 minutes. If you travel together with somebody, you should by a "Schleswig-Holstein" ticket, which costs 28EUR plus 3EUR for every extra person (plus 2EUR is you don't by it from the ticket machine but from a person at a counter), ie 31EUR for 2, 34EUR for 3, etc (up to 5 persons)). You can use all regional trains (RB, RE, AKN, IRE, NBE, NOB) in Schleswig-Holstein, Hamburg and Mecklenburg-Vorpommern, plus most of the public transport in Hamburg for the whole day (but not the bus in Kiel and no fast bus in Hamburg). You need to write your name and the total number of persons on the ticket before you start. At the train station you may get asked by random people if you have such a ticket and let them ride with you (or ask you to join them on your ticket). I recommend not to take up that offer, they may sell you fake tickets. Buy a ticket form a vending machine, and you have the cheapest way to get from Hamburg to Kiel if you travel with somebody else. The trains are not as comfortable as the ICE or IC, and there is limited space for luggage in an RB or RE, but you will have more time to enjoy the landscape of Schleswig-Holstein (and you probably need to change trains in Neumünster). Trains leave from Hamburg central station and Hamburg Dammtor at least once an hour.