There are excellent flight connections to Finland from all over the world. Finnair and SAS have scheduled flights to Helsinki from most major cities in Europe, as well as from New York, San Francisco, Cairo, Bangkok, Singapore, Beijing, Sydney and Tokyo. Twenty-two other international airlines offer regular flights to Helsinki. There are no departure taxes when leaving Finland.
Land crossings into Finland from Sweden and Norway are hassle-free, serviced by frequent buses and trains. Land crossings from Russia are a little more problematic, but border crossings are becoming more relaxed all the time. If you stick to the main tourist corridors (eg. Helsinki-St Petersburg) you won't have any troubles, but make sure you have a Russian visa before you roll up at the frontier. The Trans-Siberian Railway connects Europe to Asia, although its popularity has declined in recent years due to the general state of chaos in Russia. You can buy a ticket in Helsinki for the Chinese border via Moscow. Beware of sharks offering discounted tickets on this service; it's almost certain you'll be ripped off.
Baltic ferries run from Sweden, Estonia and Germany to Helsinki and Turku. The ferries are impressive seagoing craft and have been compared to hotels and shopping plazas; they actually make more money from duty-free shops than they do from passenger tickets! If you want to gamble while you sail, there is also a casino boat that goes from Umeå, Sweden to Vaasa.