How should I install HPLIP binary/proprietary plugin/driver for my HP printer or printer+scanner?

For HPLIP from APT make sure that you have installed hplip-gui package. If it is not installed - install it with:

sudo apt-get install hplip-gui

According to official HP documentation we need to have HPLIP installed and then install binary plugin as follows.

To install the plug-in using the GUI you can follow these procedures:

1. Launch a command-line window and enter:

   hp-setup
   

2. Select your connection type and click "Next".

3. Select your printer from "Selected Devices" list and click "Next".

4. Enter your root password when prompted and click "Next".

5. Use the recommended installation method and click "Next".

6. Check the box to accept with the "Driver Plug-In License Agreement" and click "Next".

7. Finish the installation of the printer as normal, however you may be prompted to re-enter your user name and password.

If you run into any additional install problems you can go to our known issues page and check for a solution.

For advanced users who wish to install HPLIP components manually, a utility exists (hp-plugin) which will download and install the plugin file, but it does not also install a printer queue like hp-setup does.

Do the following:

1. Launch a command-line window and enter:

   hp-plugin
   

2. Follow the directions above for navigating the GUI but remember that the printer queue will not be installed through this process.

As a complement to the main answer:

hp-setup

There is also

hp-toolbox

(it may need refreshing the printers list in order to show all info)

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The scanner doesn't seem accessible though the above GUIs, so Simple Scan or alternatives are needed. -- There is Skanlite (cannot save to pdf), Xsane (I find it a bit complicated and oldish, but has a lot of options), VueScan (non-free).