I’m in the UK and I worked for myself for a while - it is more complicated if you work another job too. You can pay tax on your Graphic Design work by Self Assessment. You need to register on the Government website to start but there is a lot of help if, like me, you don’t know what you’re doing.
Once a year you fill in an online form with details of expenditure, invoiced income and so on. I’m pretty sure the online form has a bit to declare other income including PAYE so that should be straightforward.
One thing I noticed is Annual Investment Allowance. You declare any expenditure on equipment (like a computer) and this comes off the top of the tax you have to pay. In addition to this you can make allowance for depreciation of capital assets (like a computer) over 4 years and this comes off the tax you have to pay. In effect, the tax man is paying for your computer twice. All legal.
I used to look forward to doing my Self Assessment every year because they always ended up owing me a couple grand.