All you need is doing business, f.e. be partner in any ltd co. even if the Co. isn´t active. You have to pay yourself health&social insurance (some 150 euro I guess - this number use to be generated from minimum wage, now some 270 eur).
Friend of mine from Novi Sad who´s foreign Slovak has temporary residence in my house, he got permit to stay and can now freely travel in the Schengen area. If you have some slovak ancestors by any chance this could make it easier, as a foreign slovak all you need to get permit to stay is to have slovak adress.
if you get a job in Slovakia, permanent residency visa are no problem from what I know.
Month or two ago, European commision recommended to abandon EU visa requirements for serbian and macedonian citizens. If it doesn´t happen soon let me know I can send you invitation letter and you get single entry visa in Beograd´s slovakian embassy. When in Slovakia, I could provide you with home adress you need for temporary residency visa.
Temporary residence visa means you don´t need visas for travelling in Schengen area. After graduating the school, you can come and find yourselves a proper job here.
My friend in Novi Sad could provide you with some usefull information regarding stay in Slovakia.