In Пикник на обочине, the city of Harmont is introduced and it is written as Хармонтского. After a couple of days of practicing the Russian alphabet, it was easy to decipher the "Kharmont" part, but whenever I put this word into any translator, the answer is simply "Harmont" and the "ского" part is never addressed. I tried to look it up online, but this is probably a grammar thing that I haven't gotten to yet.
Yes, it's a grammar thing.
специальный корреспондент Хармонтского радио
means "special correspondent of Harmont radio".
In this sentence, "Harmont" is:
A noun adjunct (a noun modifying another noun, "radio", like an adjective would)
Part of a larger phrase, "of Harmont radio", which itself is a complement to "a special correspondent".
In English, these relationships are conveyed analytically, using prepositions and word order.
In Russian (not always, but usually, including in this case) they're conveyed synthetically. That means that the word itself contains certain little bells and whistles that convey its role in the sentence.
-ск- is a suffix (actually, one of many) that you can add to nouns, mostly proper nouns, to turn them into adjectives. It's not unlike the English suffix "-ian", like in "Brownian motion", "Victorian era" etc., which in Russian are броуновское движение and викторианская эпоха, respectively.
-ого is a marker of the genitive case in singular masculine and neuter adjectives. The case system is a feature in some languages where nominals (nouns, adjectives, pronouns etc.) change their endings according to their grammatical role.
In English, it only works for pronouns: the basic form is "we", but you have to change it to "us", "our", "ours" sometimes. In Russian, German and many other languages, you have to do it to all nouns and adjectives.
So Хармонтского is a special form of the word Хармонт which you need to use to convey its role in this particular sentence.