There are a number of places that offer Finnish language courses at all levels, though as with learning any language, success is often based on the amount of work you put in outside of the classroom.
The best place to enrol in a language course in Finland is probably at an educational institution. Prices are often fairly reasonable and students of that particular institution are often entitled to subsidised or even free admission to language courses.
You should contact the institution directly to find out details regarding the course, and ensure that you enrol well in advance. As a general guide, most courses provided by educational institutions begin in August or September, then again at the start of the following term in January.
The Finnish National Agency for Education can provide you with further information regarding Finnish language teaching in schools for students with a foreign background.
Language training for the unemployed is provided free of charge through an Employment Agency (työvoimatoimisto). Information in Finnish on such courses can be found on the website of the Labour Administration (työhallinto).
Online language courses are a fairly popular option in Finland. They are cheap, easy and you can learn at your own convenience. Some websites even have an audio feature that allows you to practice your pronunciation. Learning online is not always the most thorough way of learning a language, though it will definitely act as a decent foundation to build on. Search around on the Internet to find a language site that suits you – there are plenty of them out there!
From language books to podcasts to online courses, literally hundreds of companies dedicate themselves to helping people write and speak Finnish.
Unfortunately, despite any of their claims, there are no easy tricks or shortcuts to successful learning. The real key is simply to immerse oneself in the language and practice, practice, practice. Read newspapers and magazines. Watch television and films. Listen to the radio. Chat with neighbours. Constant exposure to Finnish is a necessity. The more you place yourself in situations where your native language cannot be used as a crutch, the quicker you will learn Finnish.
Television is probably the quickest way to increase your level of listening comprehension. It's free and you are guaranteed to hear people speaking naturally (and fast). Don't expect to understand everything, especially things like chat-shows, as this is even difficult for people with a good level of Finnish. The more you listen and watch, the quicker you will find yourself picking up words and phrases. You will be surprised how quickly and how painlessly you will learn this way.