Moving Steam Games

The best way to do this is to have Steam installed, make backup files of the games via Steam, then restore them via Steam.
Otherwise, I think there are a few folders you need to move along with your games so that a direct transfer to your new laptop works, and even then, you'll have some broken games.

Most of the time the savegames are somewhere in your home folder and not in the common folder.

I've moved the whole old 'common' folder (the one you found in steam > steamapps) in the new steamappas folder and installed the games

That should work. I do it often to move games between machines. I copy the game I want to the new common folder then install from Steam and it finds the files.

Your save files will be in various locations if not in the cloud though.

Save files can be literally anywhere; the game folder, My Documents, and Appdata are some of the most common locations, but by no means are they the only ones. And they're not always clearly named--sometimes I find a weird folder I've never seen before, start worrying about malware, then check it out and oh, it's just a game save. I'd suggest Googling the name of the game plus "save game location" for any that you can't find quickly--it'll save you a lot of time searching.

I really wish I could make a single folder somewhere and say, "Okay, listen up, games! Don't go throwing your trash all over my computer. This is where you save." But nope, they have to clutter the place up. T_T

Good luck!

You might have to pay someone at a computer fix shop to get those files for you. I think some steam games are saved on cloud? But if you can't get the files from your old laptop then maybe try to get someone at one of those computer store to see if they can do it.

Moving a Steam Installation and Games

The following instructions are a simple way to move your Steam installation along with your games:

1. Exit the Steam client application.
2. Browse to the Steam installation folder for the Steam installation you would like to move (C:\Program Files\Steam by default).
3. Delete all of the files and folders except the SteamApps & Userdata folders and Steam.exe
4. Cut and paste the whole Steam folder to the new location, for example: D:\Games\Steam\
5. Launch Steam and log into your account.

Steam will briefly update and then you will be logged into your account. For installed games, verify your game cache files and you will be ready to play. All future game content will be downloaded to the new folder (D:\Games\Steam\SteamApps\ in this example)

Edit:
Games which support Steam Cloud saving are stored in the Userdata directory. For other games, they could be somewhere in your Documents directory, or buried in your appdata directory (which is hidden by default). Consult the PC Gaming Wiki to find out where a particular game stores its save files.

For games which supports Steam Cloud, just ignore whatever on your old hard disk and just reinstall the game

For games which do not, there are 3 common places where most of your save files are in on Windows:

• In the Steam installation folder Steam\steamapps\common
• In your user folder, specifically:
  • Users\<UserName>\AppData (in any of the 3 folder in AppData)
  • Users\<UserName>\Documents
• In the Registry HKCU\Software\

Most games using Unreal engine saves in Registry, from my experience

You can use PCGamingWiki to check where the save files are stored,
ex: http://pcgamingwiki.com/wiki/Civilization_V#Save_game_data_location

If you don't know where the save location is for a specific game, you can usually find it on pcgamingwiki. For example, here is their page for Hollow Knight.

I have found saves in:

• C:\Users\ your username \AppData. Then in one of the 3 folders.
• Somewhere in the main steam folder for the game
• C:\Users\ your username \Documents
• C:\Users\ your username \Documents\My Games
• In the registry in: HKEY_CURRENT_USER\Software

For the registry saves, you can right click on the folder for the game and click export, name it whatever you want. When you want to load the save, I think the best way is to delete the folder from the registry if it already exists and then just double click the file you created when you backed it up and it will recreate the folder. Messing with the registry can cause problems if you don't do it right, so experiment at your own risk :)