We've got a whole article diving into exactly those kinds of details right here: Testing, Testing...
Basically, they'll usually start with asking a bit about your sexual history things like how many people you've had sex with in the last 3/6 months, the kinds of sex you had and whether you are practicing safer sex. Then they'll do the actual tests which - depending on where you go and what they're testing for - can include a urine and blood sample, and yes, sometimes a urethral swab which can be uncomfortable.
Generally, if you have no symptoms the person doing the checkup won't ask you to undress, unless they are doing a swab. Many places now do urine samples instead now, so you can always ask when you're booking an appointment what to expect. In some places, the local sexual health services are sending out free DIY tests in the posts if you don't have symptoms.
Is there a particular risk you are concerned about?