Due to the system of banks being bound by a mix of laws and their own wild nature, the cards issued here differ in some ways from what may be normal elsewhere.
First off, the 4-digit card, where the remaining 12 digits are simply invisible to you, so you can't use it to shop online. To do that, you have to open a separate account and get a separate card, but then that account has some limitations - you're supposed to move money to it only when you need it.
Your magnetic stripe card may not work here. While most of readers still have the readers, and the sweep generally works, nobody's using it and if it got dirty or malfunctioning, nobody will notice. All the cards I've seen here have the embedded chip, and they get inserted into the reader, where the chips contacts touch those in the reader and thus exchange information. So if your cards don't have the chip, make sure you bring some cash - exchange offices are on every corner.
Update 2016: Now that the older card readers are being replaced, the new ones don't even have the stripe reader. It may happen that your card won't work in many places if it has the stripe only and doesn't have the embedded chip. Specially at bankomat - they all work by chip only. Beware.
Update 2016.2: Just signing your bill on your debit card won't work anymore. The software now demands that you type the PIN. Appealing to the good graces of the clerk doesn't help, you must enter the PIN or the payment won't go.
Update 2016.3: for you with foreign cards, beware. The exchange rate may not be what you expect. I've seen the rate of 113RSD=1€ when the official exchange rate was 123.40. On a bankomat you at least get a warning; on a bar tab you only get a surprise.