Am i right about Bitcoin?

So if im right with bitcoin it works like this that basically the Blockchain is just a long history of bitcoin Adresses trading botcoin to each other, so when im buying Bitcoin im giving the Seller my fiat currency so that he trades it to me. Then the Miners look throught the Blockchaing to check if the Seller can even send me his bitcoins. Then if they verified it the trade gets added to the Blockchain and now my adress got the trade in the history. Also the Adresses got a Public and a private part which I need to send bitcoins. Adresse dont get created but they get picked at random from a pool of adresses. So to use my bitcoins i bassicly just need my Public and my Private key right?