I don't know much about Serbian food, more Greek. And it may be that you don't want to stress out about your food while you're on vacation, especially if it's going to be a problem culturally. But in the summer I bet there will be a lot of nice vegetables and fruits, and healthy traditional foods. If you can maximize those and minimize the higher calorie things (fats, cheeses, pastries) that will help you control calories without being really detailed in your recording. Where I live none of the restaurants are chains or post calories anywhere, so that's what I do when I go out, and after the meal I do a rough estimate based on the size of the portions. It's really handy to have a visual idea of portion sizes in general for this reason.
You could try learning portion sizes for common types of food, like bread, poultry, beef, pork, cheese before you go, and use it to help keep general track over the day. One portion of cheese is an inch cube or Kraft singles type slice, for instance, and you could count about 120 cal. (Some cheese will be a bit lower calorie, some higher but in the long run it will work out to the average.) One tablespoon (half a golf ball) of a cream or butter-based sauce, figure 80 cal. One cup of rice or pasta (that's a tennis ball) = 200 cal. One portion of chicken = a deck of cards = roughly 100 cal. (Technically 86 cal, but I prefer to do my addition with round numbers, and it takes account of the oil probably used in cooking, smaller/bigger etc.)
Here are some handy sites for doing portion visual estimates:
Visual cues for portion control
WebMD cutout wallet guide to portions
You could write out a little sheet based on one of these, with general calorie equivalents, that would fit in your purse, and you'd probably learn them pretty fast.
Or if you don't want to carry a notebook or book but still want to keep a detailed record, do you have a phone or electronic device that you could put a calorie counting app on? There are some that come with a database so you can use them offline (like LoseIt, which I use).
Something else to practice beforehand: if your relatives are anything like mine, you might spend days going from house to house being offered a series of pastries and cakes, and not wanting to give offense by saying no. ("I'm on a diet" is not an acceptable excuse in some families.) You could try saying "That looks so good, but I'm so full from Auntie Bertha's and Cousin Fred's cake," or say "I'm just going to try a small piece." and then quietly leave the plate in a corner when you get distracted playing with the toddler cousins.
Have fun and good luck!
ETA: Just found this Mayo Clinic slide show that includes pictures of portion sizes with calories:
Mayo Clinic