Opening a bank account is easy but chose the bank most suited for your typical transactions. Once you have a Slovak ID card, even if it is just the 5 year temporary residence type card you can open a bank account.
VUB is the most expensive account I know about but they do offer some discounts if you work at particular companies. They have plenty of branches and internet banking is also easy with them. They also offer transfer services to other countries and will accept cheques from international locations.
If you want a free bank account, i.e. no monthly charges with some transaction limits then consider MBank
They allow you to use any cash machine and the cards are visa with no charges for using in Europe. The internet banking is easy but only in Slovak, again a reminder that they do have a few limits.
Do you need additional information or help, if yes contact me and I will see what I can do to help you.