If you are from outside the EU/Schengen you are allowed to stay for up to 90 days in 180 as a tourist. Whether you need to get a visa for this, or can just turn up at the border, get your passport stamped and get the 90 days by default depends on your nationality.
See this article for details of who does or does not need a visa for the first 90 days: https://en.wikipedia.org/wiki/Visa_policy_of_the_Schengen_Area
If you want to stay for more than 90 days (and are not a citizen of an EU or Schengen country) you need to apply for a residents visa. You apply for a visa to the country you will be in for most of the time. Such visas are issued by the individual countries, and each has different rules. But, once you have a visa, you can visit (but not live or work in) all other Schengen countries. Since there are no border controls between Schengen countries, they could not prevent you going from country to country.
You need to decide which country will be your base after your 90 days are up, and apply NOW to their embassy. You cannot apply for an extension of your 90 days once you in the country. You will have to provide proof of adequate funds, probably medical cover and other checks. It will be easier if you are only applying for a residence visa, not a work visa. The application process could take many weeks.
There are posts on this forum about people who have done this in various countries.
