I learned how to do this from watching Brad Beckett's video that can be viewed here. Thanks for your help Brad! Before I watched this I didn't know it was possible for foreigners to obtain personalized transport passes in Beograd. The personalized pass with your picture on it is necessary to buy 14 and 30 day unlimited ride passes for public transport.
Throughout my travels I have learned I prefer this method of paying for public transport as I feel it encourages me to explore more, better learn the system, and "get my moneys worth". I also won't think twice about hopping a bus for one stop in a sudden downpour, but would likely walk through it to save a little money without the pass.
In my opinion part of being a good traveler is respecting the laws and customs of the places we visit. Perhaps it is possible to avoid paying, but I would advise against such a path. When traveling on public transport in Belgrade, one will rarely see people interact with the validation devices. I am told the locals don't bother if they have a monthly pass, and there are often plain clothes people riding transport checking to ensure people have paid. Fines are expected to be paid on the spot, the police can become involved, which will like result in additional fines (or "tips" depending on what part of the world you are from). All of this seems quite a bit of hassle to save 89 RSD ($0.84 / €0.76) per trip.
Of the places that might house a transport office, I would never think to look for one tucked inside a corner of a department store. In Belgrade, things like this are known to happen. Here is how you can obtain your very own transport card.
First, be sure to bring your passport and make your way to the Beograđanka building, located here. The address is Kralja Milana 5, Beograd 1100. There is an underground Idea grocery store entrance and a water fountain in front. Make your way inside and go up two floors (Americans would count it as 3rd floor and everyone else 2nd) either by escalator or stairs. In one corner you will notice an area with chairs for waiting and several desks. There is a tablet terminal at the entrance with three buttons on it. Tap the one on the top and a ticket with a number will come out of the machine. Once your number is called, go to the desk and tell them you would like a personalized bus card, the woman that helped us spoke excellent English and was very kind. The cost of the card is 250 RSD ($2.35 / €2.12), it is valid for three years from the date of issue.
After obtaining the card, it now needs to be given credit, which, of course cannot be accomplished in the same place. Instead, make your way to any Moj Kiosk and ask if they speak English. Many of the clerks do not, so either use google translate to show them what you want or use this -
Здраво, пропусницу за тридесет дана молим
Hello, thirty day bus pass please
A 30 day is 3275 RSD ($30.76 / €27.79)
A 14 day pass is 1830 RSD ($ 17.19 / €15.13)
Finish the transaction with a Hvala (Thank you, pronounced vah-la)
The woman that helped me spoke no English and requested cash, I'm not sure if cash is always necessary or there was a miscommunication. Hopefully the next time I charge the cards I will be able to find out.