VAT is handled by shops for you.
Council tax (local tax based on property value) you are sent a bill based on the value of your property. There are various exemptions and deductions for exceptional cases, but otherwise you just pay the amount.
Income taxes are deducted by your employer from your paycheck under a system called PAYE (pay as you earn). It works basically perfectly if you have one job, and can even cope with multiple jobs, variable income, etc.
If you have an issue with incorrect income tax, there is a friendly call centre who are generally knowledgeable and can fix things with minimal effort on your part; and they also handle some common exceptions (e.g. dividend income under £10,000) over the phone rather than via a form.