First, they ask you a few questions about your sexual history (eg number of partners, whether they were casual or steady partners, whether sex was protected or unprotected).
Then, they examine the genital area for any visible STIs like herpes or warts. They take swabs from the penis, and if appropriate, from the rectum and throat. They also take blood to test for HIV, syphilis and Hepatitis B and C. These are all serious illnesses and can't be checked for in any other way at this stage.
For most STIs, you should only wait a week from the last time you had sex with your boyfriend before you get tested. For HIV, although you can be tested now, you will be advised to get tested again in three months' time to rule out the possibility of infection.
You would be surprised at how many people dread needles and swabs. The staff at GUM clinics know this and do their best to make patients feel more at ease. Tell them how you feel about needles, but please do not refuse a blood test as it's so much better to know you're STI-free than to worry about it for the rest of your life.
Feel free to ask anything else you want. Good luck!