The procedures concerning third country nationals entering Slovakia and the EU are in general more demanding than those concerning EU citizens.
The best starting point for the specific information in your particular case is the Slovak embassy in your country.
To get the picture about the procedures when entering Slovakia and to be ready even before contacting the embassy, you can check following sections:
Staying in Slovakia for less than 90 days
Staying in Slovakia for more than 90 days
Renewal and Change of the residence permit
Duties after arrival in Slovakia
Bringing family members with you
Changes in the student and research migration policies as of May 2018
Do you need a document to be apostilled or superlegalised? Find out what it means and what you are supposed to do: Authentication and translation of documents
Do you intend to work in Slovakia? Find more information on Slovak conditions of employment in the category: Employment