Are people with schizophrenia aware that they have schizophrenia, especially at the early stages?

Probably not but. They may sense something is wrong and tend to hide it and not talk about it.The choice to not talk may be their belief everyone feels or experiences what they do. Eventually the impact is severe enough others notice and ask questions. Hiding it becomes almost impossible but denying it, consciously or unconsciously even then is common. Sadly some are never aware, hampering treatment severely.
Some Awareness. There may be some awareness, but it not unusual to have only a vague idea of discomfort or turmoil, and not really fully understand what's taking place.
Discomfort. In addition to dr. Holmes answer, imagine how hard it would be to ignore the evidence of your own senses.
There is a problem. They know something is wrong many use drugs & alcohol to act like they are in control & causing themselves to hallucinate & act crazy. Parents initially may think they are chemically dependent & only later see they have a serious mental illness.