By air
If you want to get to Lapland by air you can choose a flight from either of the companies Finnair or Norwegian from Helsinki. There are several daily flights from Helsinki and the flight takes only about an hour. Occasionally there are good offers on flight tickets. Just check out the home sites of the companies. After arriving in Rovaniemi there will be Airport Taxis anxiously waiting to bring the visitors to the city center at the fair price of 12 euro.
By train
The railway company VR travel several times a day from Helsinki to Rovaniemi. The trip takes approx. 12 hours. I think the best is to travel by night as you sleep really good in the comfortable beds in the couches. Even on train travel you can find good prices occasionally. Keep your eyes open and check the home site of VR; the railway company of Finland.
By bus
There are also weekly bus transports to Rovaniemi from several larger cities in the South of Finland. The company OnniBus has really cheap prices and drives daily from Rovaniemi to Helsinki. The transport happens during the night and you arrive to the goal in the next morning.
By car
You can of course also choose to drive your own car to Rovaniemi. There are several main roads leading to the capital of Lapland from the south of Finland. Road nr 4 is the biggest of them. If you arrive by plane to the airport you can also rent a car to transport yourself around in Rovaniemi and Lapland for the period you plan to stay.
Depending on where you live, you have to plan you trip to Finland by yourself. You can of course choose an all in one trip with all planned for you. Many young people want to plan their own trip, though. Some trips are also made without so much planning, more like seizing the day and plan after that. I am not going to give you a throughout description on how to get to Finland from your home country. This blog is a guide to how to experience Lapland and Rovaniemi as you have finally arrived here.
From this link you find a very useful Backpacker guide for Europe, that could help you a lot about how to travel in Europe and also in Finland: