Depending on airports as different airport will have different process... There are mainly 2 different type of situation depending on the type of airports...
1) Airport that have departure/arrival on same level (eg. Singapore)... In most case, you will need to do nothing... Upon your arrival at the airport, just look around for flight information for the connection flights... If you require to check-in again, there will be transfer desk inside the terminal to assist you... And to transit between terminal, there would be bus/train connection depending on airports...
2) Airport with departure/arrival at different level (eg. Hong Kong)... In most case, you will need to follow the sign to your nearest transfer area where you can get check-in on transfer desk if necessary... If not, you can pass through security located in the transfer area which will lead you back to the departure level... At which time, you can just look for your flight information and have a shop around waiting for your connection...
3) Airport with multi-terminal (eg. London Heathrow)... In most case, you will have to follow the sign for flight connection to a centralised area where connect can be made between terminals... For airports with more than 2 terminals, that would means some terminal will require passenger to transfer by bus/train to flight connection... After you proceed security check, you will arrive at the transfer area where you can get assistance for connection flights if necessary... Otherwise, you will then be lead to catch bus/train/walk to the relevant terminal that your connection flight will depart...
Of course, the procedure will not only vary between the type of airport, but also vary between airports as process may vary... So if you find signs being a little confusing, just ask for assistance from the ground staff and they would be happy to show you the way to where you need to be...