Best place to purchase coins

Welcome to America and CT. You have already received some great advice from well know members, so I can't add much, but I would research and find a local, or online, coin club that specializes in the coins you are most interested in. Obtain some books on the same and educate ourself.
"What’s the best advice and place to purchase coins with the really possibility of searching through authentic, not fake USA coins to find the rare one?" Looking for "rare" coins to get rich with is a newcomers dream, but I think most of the members here will back me when I say, "Don't plan on getting rich from coin collecting." Good luck.

 

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The 1833 coin in the first set of photos is commonly referred to as a Capped Bust Half Dime. That appears to be nice coin grading Fine. There is something going on with the second S in STATES. It appears to either be a die chip, which obliterates the S, or a flattened area at the S, as a result of post mint damage (PMD). The photo is not clear enough to determine in the image that I viewed.

It may interest you that the Capped Bust design was also used for dimes, quarters and half dollars of that time period. No US dollars were minted from 1805 to 1839.

Welcome to CT!