I searched for Apple, Microsoft, etc. company stocks but found that there are also some other things like AAPL NE, AAPL MX, etc. What do they mean?
It is not possible to answer in general without knowing exactly what "other things" you have found. There's also the problem that stock ticker symbols are not standardized. ISIN and CUSIP numbers are standardized, so that's probably what you would use to compare the equivalence of two ticker symbols.
I can answer for the examples you have given: AAPL NE is a "Canadian Depositary Receipt" traded on Canada's NEO Exchange. AAPL MX are shares of Apple Inc. traded on the Mexican Stock Exchange's Global Market ("SIC").
I would also like to know how many stock markets there are in the US.
There are many stock exchanges in the US. Refer to Can US stocks list on one stock exchange but trade on other US stock exchanges?
... must we join each market separately to have access to all company stocks?
People buy and sell shares through stock brokers. If your stock broker provides access to many markets, then you will have access to many markets through that stock broker. You will have to ask your stock broker about the markets that they offer, since each stock brokerage firm is different.