So, I initially came as an au pair in Finland as an EU Citizen. I decided that it wasn't for me so I came to Helsinki to work as a English teacher. I did the EU Right of Residency and got the Personal Number while I was an au pair, however I also showed that I had enough funds to support myself if I lived without being an au pair. Making the change from one job to another job is easy as long as you are an EU citizen and have all the documentation.
However, I also have a friend who is an au pair and she is not an EU citizen, and if she wishes to stay after her au pairing term is done, then she will need to either enroll in study or get a permanent work contract so that she can get another residence permit. The permit she is currently on is one for an au pair, if the circumstances change, she will have to reapply for a new permit or leave Finland.
Keep in mind that au pairing is not like any other job, you live with the family 24/7 and if you don't like children, it will be a very hard situation. I will also advise that if you do decide to become one, then choose bigger cities instead of small towns as you will have a better experience and meet new people.
As far as getting a position, I applied online through
http://www.newaupair.com . There are several reputable au pairing sites out there, you don't necessary need to go through an agency. That does not always mean you get a nice family. Good luck!
By the way for everyone else, au pairing in not supposed to be a real job. It is supposed to be a cultural experience and encourage young people to experience a country while "offering" a few hours a day for light house cleaning and some babysitting. All your meals are included, so is your room and board and usually on outings with the family, they pay for everything. The pocket money that is given is a small stipend to help with travel costs and any other things that the au pair might want like new shoes, extra food, etc.