Kirkenes is at the very end of the all-through-Norway (and half of Europe) highway E6. You can also enter by road from northern Finland (via Rovaniemi about 7 hours), and northwestern Russia (via Murmansk about 3 hours + border formalities). If you want to come from southern Norway by car, cutting through Sweden and Finland is actually faster.
Long distance buses , with connections to most of Norway.
Buses to Finland (summer only): 
Due to its distance from the rest of the civilized world, most people choose to arrive and depart by plane, at Høybuktmoen airport (IATA: KKN) (ICAO: ENKR) some 15 km outside town. Daily flights to Oslo, Tromsø and others by Norwegian , SAS and Wideroe . Also flights to Russia and Germany (summer only). Most flights have a bus to town waiting outside, 85 NOK, credit cards accepted. A taxi will set you back some NOK 300.
On certain occasions it is considerably cheaper to fly into Ivalo in Finland, some 240 km away.
Although there is a railroad here, it is for ore only. Real adventurous spirits should be aware that you can connect to the russian network in the border town of Nikel, travelling as far as Moscow, Vladivostok, Ulan Bator and Beijing.
Local buses . Long distance busses leave outside the main shopping center in the middle of town. Some busses only take cash, however there is an ATM inside the shopping center.  is a good website to plan your travels.
If you plan on crossing the border into Russia, you will probably need a visa. In theory, this should be obtainable at the Russian Consulate in Kirkenes, but you might find it worthwhile to pay any local travel agency a little extra for the job.
If you are heading for Finland, and feel that you need some euros, there is an ATM at the DnB bank that will hand them over.
Go up the Pasvik valley for quite a change of fauna. Here you find the last bit of the enormous Russian Taiga, a pine forest that stretches all the way to Siberia. At the southern end of the valley (105 km of paved road, 15 km of dirt, then 1h15m walking) is the Tri-Border Mark, where Norway, Russia and Finland meets. Heavy fines for trespassing into Russia!
At Grense Jakobselv, the open Barents Sea washes a beautiful sandy shore. Never gets above 8 celsius, though! You can also peek at the "King Oscar II" church.
The highest peak around is "Øretoppen", at 466 metres. It is a 2 hour hike along a marked path from Ropelv, where buses go. Magnificent views in a landscape of large stones spread upon the hills.