You can likely cut 60% of your "that's" and your story will be unaffected. Sometimes, you do need to add "that" here and there for clarification, but not always. And sometimes it's just plain incorrect.
E.g: The jacket was the coolest one that he'd ever owned.
Vs. The jacket was the coolest one he'd ever owned.
In other cases, you might do well to substitute "that" with "which." Though, if you're doing this, make sure you do it properly. That change can often alter the meaning of your sentence. That can be for the better, though.
E.g: The vandalism that read "Bad Wolf" made Rose nervous.
Vs. The vandalism, which read "Bad Wolf," made Rose nervous.