if I install a Steam game on another computer, will my progress from my first computer be available immediately? Or is this game dependent? If not, what’s the best way to transfer the progress?
If a Steam game supports Steam Cloud, savegames are automatically synced with Steam servers and are available on other computers if you log in with your Steam account. It does not necessarily store all your savegames, e.g. with Civilization V you have 10 savegame slots on Steam Cloud, so when you save a game you have to choose to save there, else the savegame will only be local.
You can use this search to find a list of Steam Cloud games.
For games that don’t support this you could try synchronization with online storage providers like Dropbox, I never tried this myself but it should be possible. Most games have their savegame data somewhere in you personal home directory, often inside the “my games” folder.