If you want to stay in the Czech Republic for a longer period of time, you need to obtain long term visa/residency. Every type of visa has its own purpose of stay. There are different types of the purpose of stay.
You can apply for this purpose of stay if your university is accredited or if you are participating in official language course preparations for Studying in the Czech language. You will get a long-term visa for one year. After the first extension, you will upgrade it to long-term residency which is valid for 2 years.
If you will be studying in an unaccredited study program at a public or private higher education institution
3. Scientific research
If you will work in the Czech Republic your purpose of stay is employment and you will get a long-term residency for two years. There are types of residencies with the purpose of staying Employment.
An employee card is a long-term residency with classic employment at least with a minimum salary without any job type qualifications.
Blue Card is The higher equivalent of a Czech employee card. Higher requirements such as specific degree and monthly salary.
If you want to open a company in the Czech Republic or be self-employed (freelancer), your purpose of stay will be business. You will get a long-term visa for one year. After the first extension, you can upgrade it to long-term residency which is valid for 2 years.
6. Family reunification
For this type of visa, you need to present proof of relationship (marriage or birth certificate)
7. Job seeker visa
A job seeker visa is a long-term residency with the purpose of starting a business or finding full-time employment.
Residency permit valid for up to 9 months. All students graduating through accredited programs can apply for this permit. This permit can be converted to any other long-term residence permit including the business one.
Changing the purpose of the stay
It is not possible to change the purpose of stay to the business unless you have been staying for more than 5 years in the Czech Republic.
An exception is a family reunification visa, after 3 years of holding this type of residency you can change your purpose of stay.
Here you can find complete information