There is a finite number of universities in Finland. Disregard universities of applied sciences (ammattikorkeakoulu); they are teaching institutues with some R&D and do not offer PhD positions. Wikipedia contains a list of proper universities: https://en.wikipedia.org/wiki/List_of_universities_in_Finland . I recommend starting with technical universities, including Aalto university.
Go through those universities and check if they have faculty in relevant areas. Follow the open position announcements in all of them. Apply when you find something relevant.
Also, you should contact relevant faculty and ask them about possibilities and vacancies. If your university has contacts to Finland, use those.
I recommend starting to apply when you know you will receive your master's thesis soon.
Recruitment is university-dependent. There might be two open calls, one ending around now (maybe has ended already) and before new year, or there might be open calls whenever there is funding. I do not know if it is possible to hire a PhD student without an open call; I would assume yes, but this might very well be university-specific.
There is typically an online form that you fill. It asks for studies, employment history, and maybe language skills and publications (if any) and so forth. There is typically room for a freeform application or motivation. Attach CV, maybe list of publications, maybe a cover letter or letter of motivation etc. I would not expect anyone to ask for letters of recommendation, but they might ask for people who can recommend you. I would mention such people in the CV.
On CV (no photo included, typically): http://www.tenk.fi/en/template-researchers-curriculum-vitae , http://www.aka.fi/en/funding/how-to-apply/appendices-required/curriculum-vitae-guidelines/ . These are guidelines for making a general academic CV. Leave off the empty categories.
Most PhD students get paid reasonable, but not large, wages. The program is typically four years with 60 ECTS credits of coursework. Some are funded entirely or partially by grant money they bring in themselves. I cannot give specific advice on grants; please ask someone you would like to study under.
Random vocabulary:
- PhD student: jatko-opiskelija, tohtoriopiskelija, tohtorikoulutettava
- grant: apuraha
- university: yliopisto
Educated people and young people usually speak English, but I would still recommend learning at least a little bit of Finnish. It sends (a weakish) signal of commitment and interest if you can write on your CV that you know or are learning basics of Finnish, and will certainly make life and navigation easier.