How Bitcoin wallets and it's transfer process work? I am very new on cryptocurrencies, and like to understand it's transfer process. I think if I want to buy Bitcoin, I must download a Bitcoin wallet, then find a person who has Bitcoin. Then he/she must transfer that bitcoin from his/her wallet to my wallet. Then I will pay the price either in cash or with credit card to him/her bank account. 

After that I can see this Bitcoin in my wallet. OK? But I have some questions:


How many wallets a person can download and use? 
If I used a wallet in the above example, then I wanted to change my wallet, should I download a new one and do a transfer from my previous wallet to new one? I mean there is no unique ID for people that they can use it in different wallets to reach one unique account?(Like our stack account that we can reach to it by Chrome, Firefox, Edge, etc, considering the browsers are wallets)
How crypto exchange websites work? If I want to buy or sell 1 Bitcoin, should I pay the money/coin to their bank_account/Bitcoin wallet, and the other person should do that too, then the website will pay it to us? I mean is it a 2-step process or buyer/seller do a direct transfer?
First, a wallet is a software application that usually manages your keys, addresses, and transactions. When someone sends you Bitcoin, it is generally sent to one of the addresses that the wallet software manages. There really is no limit to how many wallet software applications you can run, and not really a limit on addresses either (except for disk space, but that's not really a concern etc.).

Your wallet software should allow you to create new addresses that you can use to receive Bitcoin.

Exchanges work by managing your Bitcoin in their internal wallet software, and you exchange anonymously with others by placing and filling orders on the global order book. The exchange manages the Fiat (dollars) currency exchange internally as sort of an escrow.