A map of the tramvaj system can be found at: http://www.zet.hr/pdf/mrezadan.pdf.
If you take a bus from the airport to the bus station, you'll be at "Autobusni Kolodvor" (the label is between Zitnjak and Borongaj on the right hand side of the map.) You can find your way out of the front of the building, facing a big street. If you cross the street at the crosswalk and cross over the tramvaj tracks, you'll be at a tramvaj stop going in the correct direction. The most direct way to either hotel is to take a #6 tramvaj - the second large square you come to, with the second statue of a guy on a horse (Jelacic Place, or Trg Bana J. Jelacica on the map) is where the Hotel Dubrovnik is located. The tramvaj will have turned left, then right, then left again just before it gets there. Two stops onward from there is the stop where the Hotel Ilica is located.
Other combinations of tramvajs from the bus station will get you there, but this is left as an exercise to the reader.
Before taking the tramvaj, you should buy tickets from pretty much any newstand in town. You want to ask for a "tramvajska karta". (trahm-vye-skah kahr-tah.) When you get on the tramvaj, you stick the ticket into the little machine at the front or back of the car, and it stamps and dates it.
The wedding is also accessible by tramvaj. The south-bound tramvaj stop for Trg marsala Tita is right outside the building, and across the street is the Croatian National Theater, a hard-to-miss ornate yellow building on an open green. To get there from Hotel Dubrovnik, you can take streetcars number 12, 13, 14 or 17 heading west (wait on the side of the square closer to the statue, across the tramvaj tracks) for two stops. From Hotel Ilica, take any streetcar eastward for one stop (wait across the street for the hotel), then walk just around the corner to the right to the other streetcar stop and take any streetcar for another stop. When you enter through the main doors of the museum, the restaurant door will be immediately on your right.
To take the tramvaj to bert's parents house from either of the hotels, you'll want to be taking a #6 or #11 tramvaj to Crnomerec, the terminus of those lines where there is a turnaround. (The #2 tramvaj also goes to Crnomerec, but doesn't pass the location of either hotel; if you happen to be at the train station or bus station you could also catch a #2 tramvaj from there.) From Hotel Ilica, the stop is on the same side of the street as the hotel. From Hotel Dubrovnik, go to the main square and cross over the tramvaj tracks to be on the correct side for the Crnomerec direction. Get off at the turnaround at Crnomerec (everyone will get off here, it's the end of the line.) Walk in the direction the tram has been going, around the loop to the corner, and cross the street at the crosswalk. (you're basically amking a left turn across Ilica, the street the tram was going on.) You will wind up on Gjure Szaba, a street not much wider than driveway (but beware, it does have two way car traffic.)
They are at Gjure Szaba 13, a doorway marked 13 partway down the 2nd apartment building on your left, and there's a buzzer marked "Dvornik".