Correct peoples grammar
Retina, 80 74 73 71 74 64 65 62 78 72 bytes

 *([.,;!? ])(\1| )*
$1
T`l`L`[.!?].|^.
([.,;!?])(\w|')
$1 $2
[^.!?]$
$&amp;.


Try it online!

This is my first Retina answer.

-12 bytes thanks to @BusinessCat!

-3 bytes thanks to @Neil.