Is there a way to keep the character sheets within a campaign private to just the player using the character? I have some information I'd like to have on their sheets but I don't necessarily want the other players to have access to the information.

Is this possible?

Yes, at the bottom of the Home page in the character editor, switch from Public to Private. 

A private character can't have its sheet viewed by anyone except the owner and DM, but the campaign page will still show name, level, race (not subrace), class, and subclass.

Thank you!

Aren't the Description and Notes section only visible by the player and the DM anyway - whether the sheet is private or public? Or am I wrong?

Quote from Emmber >>

Aren't the Description and Notes section only visible by the player and the DM anyway - whether the sheet is private or public? Or am I wrong?

 Correct. Extras too.

Any way to hide the race and class from other players? Trying to keep my new character as mysterious as I can to my group 

Species yes (though you won't like it), class no.  You can homebrew a copy of your species and rename it whatever you like without changing the mechanics at all.  Usually to 'Human' or something similar for characters that can pass as human, i.e. kalashtar, changelings, and the like.

Class you're stuck with.