TL;DR: for "simple" employees, pretty quick and simple. for freelancers / small businesses the complexity scales based on type and size of business: from pretty quick (for freelance licenses) to more involved (for high turnover or special corpo licenses etc. i don't really know).

Generally if you are employee and all your income comes from single employer, your taxes are done by the employer and simply withheld from your paycheck. Some are paid continuously and then balanced (excess returned, never the opposite) some time in the spring, but as employee you don't have to fill any forms. (With small exceptions -- IIRC we had to sign one "pink" form every year, I don't recall what exactly it was for.)

For long time when I was employee I also had income from stocks, so then my employer did not have to do taxes for me so I had to do it on my own (ie. I used advisor.)

If you are self-employed then it depends on the type of business license / trade and turnover: you may need to track every expense and then compute & fill form in April (this term moves to June if you use accredited tax advisor).

There are also license types where you don't need to keep expenses, you only need to report yearly income and tax based on that number. The license has a percentage number assigned and it's just assumed that your expenses was that % of your income. When I was freelancing as a programmer, I used this kind of license. It was pretty comfy for me, even though i still used advisor since I can't come close than 2m to a tax paperwork or I'll start panicking :D

Once you get past some turnover (i think it's 2 million CZK per year--about $100k) you need to use more complex system which requires you to do extra papers/taxes related to VAT, plus there's more tracking. (You need to track expenses precisely and keep the invoices/bills for some amount of years.) The rules are probably also different for different kinds of corporations, for some regulated trades, etc. For even larger businesses some of these are done half-yearly or quarterly.

Social & health insurance (both of these are mandatory, ie. they are basically taxes) is paid separately from the above (for businesses. for employees this is still done by employers) as advance payment and then balanced based on your reported income. (Then there are several small taxes like public broadcasting, land taxes or city taxes, but those are usually really easy to pay, they are either same per capita, or in case of land taxes you just report your land size and the burreau will compute it for you and send you the bill.

Of course there are also deductions based on whether you have kids, spouse, mortgage, charity gifts, etc.