Shifted print, worse during first layers

I am experiencing some shifting in the x axis. After experimenting with higher voltage on the driver (adjusting to 0.7V instead of 0.5V), printing over USB and SD, trying different slicers/hosts and adjusting the belt tension, the results still look largely the same: the raft is badly out of shape and the first layers are rather horrible. The higher layers look better, but are still leaning to one side. (see images). The axis looks to move lightly enough and doesn't stick anywhere.

What could be the reason for this? Where could I continue the troubleshooting?

Lots of possibilities described on the wiki at [reprap.org]

Slipping pulley on the stepper motor is easy to miss and has to be a possibility given the other things you've checked. I like to put a witness mark across the end of the shaft and across the face of the pulley so that I can see if they get out of alignment. Too high speeds and acceleration can also be quite subtle and only show up with certain prints, depending on the tool path generated by the slicer.

I was quite sure that my pulley is well fixed, but once you say it is hard to detect I am going to make the line test.

If that is not the problem - is there any more likely cause, from looking at the pictures?

And how to change settings in the firmware? Is there an instruction for that anywhere?

I'm afraid it's a bit of a process of elimination, you just need to methodically work through the possibilities until you find the cause. The wiki page looks daunting, but in practice several of the problems collapse down to the same solution (i.e. reducing speed and acceleration is the fix for multiple underlying causes).

How to change the firmware depends on the electronics and firmware you are using. It's very useful (almost to the point of necessity) to be able to change settings on the firmware so it's well worth figuring that out. The newer firmwares on the higher end electronics can be configured by simply editing a text file and uploading it to the board. Even the simpler boards can have many of their configuration settings saved into eeprom via g-code, but only if the firmware was setup that way when originally compiled.

James, you were right. The pulley was the problem after all, and if you hadn't mentioned how hard this is to spot, I would have thought "Meh, of course I now how to tighten a grub screw!". I also found that feedrate and acceleration settings can be changed easily over the Repetier host.

And so, after lots of silly mistakes, my son and me finally managed to print something decent today!

Great! Congratulations - now be prepared to spend silly amounts of money on filament