Is 目 suitable for daily conversations, or should I use 眼睛 (and what about 口 and 嘴巴)?
In classical Chinese, most of the words only have one character, for example, "目", "口", "道" (道路, road), "卒" (士兵, soldier)...

One of the big differences between classical Chinese and modern Chinese is that in modern Chinese, most of the one-character words are replaced by words with lots of characters (usually 2).

So both "目" and "口" are not used in spoken language, and only in some special circumstances (explained below), they are used in written language. Attention: although "目" and "口" are not commonly used individually, they appear a lot in the words, for example, "口才", "目光", "港口", "目录"...


  目 is better as radical to explain other complex characters but 眼睛 is something you can talk with real Chinese person about


This means in daily conversation, you should use 眼睛 instead of 目, but the knowledge "目 means eyes" can help you to learn other complex things, for example, when you see "睁", or "盯", you should know they are related to eye. "睁" means open (eyes), and "盯" means stare.



Another thing: if you look at "睁" or "盯", you will find there is a typical way to construct the Chinese characters. Usually one part of the character shows its meaning, and the other part shows its (potential) pronunciation. 
In "睁", the left part shows it is related to eyes, and the right part shows its pronunciation is zhēng.
In "盯", the left part shows it is related to eyes, and the right part shows its pronunciation is dīng.




  So both "目" and "口" are not used in spoken language, and only in some special circumstances, they are used in written language.


The situations to use "目" and "口" individually are extremely rare. Not only the beginners, even the common Chinese people don't use them in this way.

Usually when some writers want to make their works more comprehensive and variable, they will use "目" or "口". For example, 这时候她会有种奇怪的感觉,好像自己身上长满了嘴,张大着口,嗷嗷叫着要人喂. (Since 嘴 and 口 mean the same thing, the author uses both of them to avoid repetition.)

And also since "目" and "口" are words in classical Chinese, using these words can make the articles look more "classical".