Moving Steam to New System

Copying the steamapps folder from inside your old client to the new one is enough to move the games (as long as you are using the default download location for steam if you used another one just move that and add it as a download location in steam settings it will detect the games automatically). As for progress it would depend on the game if it uses Steam Cloud you won't have to do anything. Some games keep the progress in the game folder others in your Documents folder so make sure to check there too. Not sure if there are other locations for game saves I personally haven't seen any.

there should be a "backup and restore games" button under the "steam" drop-down in the top right, so probably backup to an external drive or sth and restore on new rig

Not sure about the settings, but I'm pretty sure the progress is (in a lot of cases at least) saved to the cloud, so if you download it on another pc, you should be able to keep on playing where you were. But what luller said won't work I think, as steam (and some games on steam) sometimes also save on other locations.

you need 3 things.
The first two things are files in your steam folder called "ssfnXXXXXXXXX". These files are important. Steam will recognize you with these files. This way you don't have to wait a week for steamguard to let you trade/buy games and stuff. IMPORTANT: Don't ever share these files. If anyone gets them it's easy to log into your account and do everything with it.
The second thing is your steamapps folder.

You don't need anything more than this. Except that some games save their settings and savefiles in a differtent location than your steamapps folder. So beware that you might lose some of it if you don't carefully check every game.

I usually do the backup using own application system. I think preferable to a clean installation to prevent problems, although these seem to be very rare.

Alright. So the idea I'm getting is that I might as well do mostly clean installs, and then go back for save files for games one at a time if Steam turns out not to have copied them for me. That works. I was hoping for a cleaner solution, but apparently Steam doesn't support one.

Saves of many games are in the cloud, you can also look for local saves in game folder (.../common/game) or in appdata folder (open up Run dialog box [Winkey + R] and type %appdata%). As somebody mentioned you can always make backup copy of game files (right click game in library - properties and look there). Copying whole folder is waste of time and space, you also copy tons of junk, but if you don't mind It's always an option.

To copy Steam games, you literally just have to cop the Steam folder.
For saves and configuration, you either move the entire Documents folder or manually sort through the whole thing.

The slight disadvantage is that this way the registry entries won't carry over. Normally it is not a big thing, but if you use any third party stuff for a game (mod manager, save editor, sometimes even a trainer), then they won't be able to find the game. (Most offer manual search, but not all programs.)

Gamesave Manager should move over most of your saves, though it may miss a few.

steam/userdata folder stores pretty much every indie game savegames, I always copied that folder when I reinstalled windows to not lose any savegames.

What's your definition of huge?

I did this, got some bugs

Isn't better just do a clean install and a backup of the games in the old system through Steam?

Wasn't there an application for this?

Safest is always to just copy the whole steam and my documents folder.

There is a program called Game Save Manager, you can export and import your saves with it.

If it was me, I would just install steam on the new PC, then replace that folder with your older steam folder including all the games.
It will copy over your settings, games, login, etc. You wont even have to wait 7 days before using Steam Market.