For most Europeans, Finland seems to be far in the north and difficult to reach compared with Central European countries. Thus it may take some effort and money to get here - but nowadays there are many budget airline companies flying between Finland and other European countries.
Helsinki is by far the most popular destination in Finland. Helsinki's international airport is actually located in the town Vantaa, about 25 kilometres from the centre of Helsinki. Thus the airport is sometimes referred to as Helsinki-Vantaa Airport.
I would not recommend you to choose Finnair for your flight to Finland unless you need intercontintental connections. If you, on the other hand, are just staying overnight between intercontintental flights you might choose to stay in some of Helsinki's airport hotels. There are several good alternatives of which I recommend especially Rantasipi Airport and Cumulus Airport Hotels.
For European flights, the tickets by competing airlines (e.g. Lufthansa and KLM) are usually a lot cheaper than Finnair's tickets. Thus I myself rarely fly using Finnair, at least if I have to pay the tickets myself.
Some of the budget airlines worth considering are:
- Air Baltic
- Wizz Air
Of these three, only Air Baltic uses the Helsinki Airport. Wizz Air flies from Turku and Ryan Air connects the Tampere Airport with Central Europe. Thus you can choose any one of these depending on the cities you plan to visit.
As far as I know, there are no real qualitative differences between these airlines -- you can simply choose whichever of these operates connections closest to you. Moreover, the train connections between these three cities are excellent and you can hop from one city to another one of these three in two hours. In fact, this is what I recommend to you: you should certainly visit each of these three cities if you are investing money to a week in Finland!