If I am interested to see the individual transactions in a certain exchange market (e.g. NYSE) for a certain stock (e.g. GOOG) - is it possible?
The NYSE, as well as all other exchanges, provide historical trade data.
Third parties often collect data and provide it for cheaper prices, but the data cannot be guaranteed to be complete or accurate.
All trades can be exported in real time from a data feed; however, some minor exchanges will not provide clearing agent information until after the close to provide for more anonymous trading. Major exchanges provide clearing agent information in their data feeds.
Interactive Brokers gives a transaction history limited by some time with the same data as above except possibly the clearing agent.