What you see on your screen is the result of your computer and monitor settings.
Mostly, your monitor settings.
The Printer is likely the most accurate part in the equation.
Start by printing some tests.
This page has some good test images:
Printer Test Images - colour and monochrome images for testing
You need to make sure the Printer is printing properly, so do a few different tests.
Test both B/W and colours.
Print an image like this and judge the colours you know (not what you see on your screen).
The trick is to adjust your monitor settings to match your Printer output/image.
There are some small changes you can make with your printer, but adjusting your monitor is a lot simpler.
This will also help you see any changes in how your printer does...say, when ink gets low.
After printing an image, you can compare it to you screen and adjust your screen colours to match.
This of course, might throw off things you've already become accustomed to.
The look and feel of a particular webpage for example.
But you'll get used to the differences.
...
As far as checking settings, print settings are "default" for every print.
You need to familiarize yourself with what is the default whenever you go to print.
That way you will get used to whatever changes you must make for each desired print.
I'm not sure you can change 'default" settings to a new set of default settings to better suit your common print jobs.
That would be nice.
But I don't know what your printer is capable of.