Since Term Store functionality (and probably most of metadata functionality) isn't available in SharePoint Foundation 2013, I couldn't find a way to search through the pictures using some sort of tagging. Thus I decided to employ something what is available already in Foundation version.
When you edit the picture, you can see 3 fields: Title, Description and Keywords like so:
It would be nice if I could make Search index terms (tags) added to the Keywords field. However, after some testing I saw that only Title is indexed and presented in search results. Although I could use my search terms in Title field, it won't be elegant.
So, is there any way to make use of Keywords entity in my case? Please note, it's a Foundation version, so there is no Enterprise Keywords functionality either (or at least I couldn't find one).
OK, so I used this kind of workaround in the end:
Now I have Title, Primary Tags and Secondary Tags indexed fields available for each picture with a total of 765 characters available.