If you marry while he is on ESTA/tourist visa, he goes back to his own country IMMEDIATELY. He cannot remain in US. ESTA (visa waiver) strictly prohibits any extension or any change of status. Marrying on a tourist visa is technically a visa violation! He goes home! If he overstays or commits any visa/ESTA violation, he goes home & applies for your immigration to his country.
Only after he returns home would you file for a spouse visa (CR-1) so that he could return to US (or he files for your immigration to Finland). You must prove you earn enough to support self, any dependents (don't get knocked up), plus spouse. You need current year paystubs + your last 3 years of US income tax returns as proof of earnings. You must also have a suitable home for you & spouse. Spouse visa typically takes about twice as long to obtain as a fiancé visa (K-1). He cannot enter US while immigration visa application is in process, but you can go to his country to visit him.
IF you two decide to marry, the proper way (& fastest) is for him to return home on time. Then you file for K-1. You must prove earnings requirements, same as for spouse visa. He will be interviewed, etc, at the US consulate nearest his home. If approved, he gets his fiancé visa, and you will have 90 days from his date of entry to marry & file for Adjustment of Status. Takes about 6 months for him to get a green card. Then he goes to Social Security Administration to apply for SSN (form SS-5) - takes about 4 weeks to get his new Social Security card. Once he has green card and Social Security card, he can start looking for work, start working, get a driver's license & everything else. You will totally support him (including getting his health insurance) until he's legal to work, finds a job, contributes to the household.