The shifters and brakes are one piece I found this old junkie bike and I want to use it. I don't have very much experience with bikes, and I noticed that that the speed shifters and the brake handles are one piece together. Is that normal? Do they normally come as one piece?

Because I was thinking of getting new brake handles but I don't want to change the shifters.
That is pretty normal on entry-level mountain bikes. Shimano claims this is to not confuse beginners.

Torney, Altus, Acera and Alivio groupsets come this way, at least by default.

However, Acera and Alivio are getting better and better with the trickle-down tecknowogy.

See here, in the upper corner "Components".