Ubuntu 18.04 not working with Intel Integrated Graphics

If I understand correctly, your first wish is to run on the IGD exclusively, then sort the remaining issues. The following steps should achieve the first goal to increasing degrees of thoroughness.

Check for improvement after each step, and stop when it works. Don't needlessly blacklist the kernel modules, as this could mean the discrete card won't be powered down. In fact, you may want to swap steps 3 and 4.

1. Configure Xorg to prefer the Intel

Create file /etc/X11/xorg.conf.d/10-intel.conf (this may require mkdir /etc/X11/xorg.conf.d), containing:

Section "OutputClass"
   Identifier "Intel"
   MatchDriver "i915"
   Driver "intel"
EndSection

2. Disable modeswitching out of IGD

Edit /etc/default/grub and add xdg.force_integrated=1 to GRUB_CMDLINE_LINUX_DEFAULT. When done, run sudo update-grub before rebooting.

3. Blacklist the nouveau kernel driver

Edit /etc/default/grub and add modprobe.blacklist=nouveau to GRUB_CMDLINE_LINUX_DEFAULT. When done, run sudo update-grub before rebooting. (For AMD blacklist radeon, amdgpu, etc..)

4. Uninstall the Xorg nouveau driver

sudo apt remove xserver-xorg-video-nouveau  # or radeon, amdgpu ...

This will trigger removal of the xserver-xorg-video-all meta-package, which by default is installed. That is alright, but you may want to keep xserver-xorg-video-intel installed (but see update below):

sudo apt remove xserver-xorg-video-all
sudo apt install xserver-xorg-video-intel

Update 2019 the package description for xserver-xorg-video-intel now recommends against installing the package. It should only be needed on old (pre 2007) hardware.

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Note: this is an edited copy of my answer to this question, which may have useful pointers for your follow-on steps, as it is about resolving issues with the discrete card.