It's a bit subtle: "it" is the 'singular, neuter, personal pronoun' and "that" is a "demonstrative pronoun".
What this means, is that you use "that" to be selective of something. So to take your examples:
"It is an amazing place." -- usually, after you've introduced the subject, you can carry on with 'it'.
"That is an amazing place." -- usually, you use it in response to somebody else, and using it for emphasis.
"It's good." -- again, you introduced it and carried on with 'it'.
"That's good." -- you're emphasizing that 'one part' is good, or in agreement with something else.
"It's a beautiful day." -- this is typical.
"That's a beautiful day." -- this probably wouldn't be said by a native speaker in the present tense, unless there's special emphasis, and it would probably be in the past tense: "That was a beautiful day."
However, whichever one you use, you will be understood, so try not to worry about it!