I have met some Christians who seemed to put a lot of emphasis on their realization of how sinful they were. Others, however, didn't seem to be stressing this point that much.
Honestly, in my personal experience, the joy of knowing the Lord, the joy of the very first prayer to Christ, and the joy of having fellowship with Christians came first. The first true realization of how sinful I am came much later. In fact, this realization is still progressing.
By looking briefly through the New Testament I can think of some examples of individuals' coming to the Lord with a considerable level of contriteness over their sinfulness, for example, the robber on the cross asking Jesus to remember him in His kingdom. On the other hand, there are also some examples of people coming to the Lord for the first time, in which this contriteness is not clearly stated, for example, the eunuch of Ethiopia in Acts 8.
So, my question is how much realization of one's own sinfulness is needed for a person to be saved?