You're right there's no specific notation to denote that a word is based on Chinese characters vs. is pure Hangul. At times, in written text, the Korean word (i.e., in Hangul) is written and then the Chinese characters for the word are written in parentheses next to the word. This is in case there could be confusion regarding the meaning of the word. I've seen this sort of thing more in academic textbooks though.
I think the best way to start recognizing Hanja themselves is to learn the first few basic ones and then use the context of the situation to figure out whether the words are Hanja or not.
Learning words in Hanja isn’t really the same as learning individual vocabulary words, such as learning the word for cat or dog or the like. You asked about "spotting" words. I guess one way of "spotting" them is more like knowing some various basic characters and their meaning and trying to figure out whether it makes sense for those characters to be part of a word. It's like seeing Hanja as being made up of distinct parts.
The best way to see it is probably by example. So, for example, you can write the words for different languages like Chinese, Japanese, and German etc. fully in Hanja as 中國語, 一本語, 獨逸語, respectively.
You can also write those words in Hangul as 중국어, 일본어, 독일어.
The common thing you probably noticed is that all three words end in the Chinese character 語 , which is expressed in Hangul as 어, and which means language.
By knowing this one character, you'll probably be able to figure out other words that you might not have seen before more easily by context. For example, 한국어 means the Korean language. 단어 means language.
So, if you hear a word with 어 in it, and from the context it seems like something related to language, it's highly likely that the word can be expressed in Hanja. I guess this is some sort of heuristic if you want to call it that. Hope it helped!