Is it common to learn individual characters before two-characters words?
I can tell you about my experience, I typically follow a mixed approach learning at the same time multiple characters words and its individual components. I learn Chinese doing lessons following the HSK exam levels (currently I am doing the HSK level 3). In every lesson, they introduce new words given by the HSK level. The order in which the words are introduced correspond approximately to their usage frequency; I believe learning the most used words and grammar structures is the optimal way for learning to communicate in Chinese (note, for example, the top 500 characters are used the 79.2% / 72.1% of times).
So my main goal is learning the words in this order. However, I find very useful for multiple character words to look for the meaning of their individual characters. Some of them are very general and abstract and they are not helpful, but in other cases they have clear meanings and help you understand and remember the multiple character word, such as 飞机 (flying machine: plane), 照相机 (machine for taking pictures: camera), 头疼 (head hurts: headache), etc.
In conclusion, if your aim is to communicate in Chinese, I would suggest to focus on learning the most frequent words. However, to understand multiple character words, it can be useful to look for the meaning of each character and on the way, sometimes learn these.