What's an STI? Same as an STD only more accurately named. Let's move on.
STI tests are important things. Condoms are the only form of contraception that reliably prevent infections and if you're moving to an alternate form then you need to get tested and so should your partner(s). In general, unsafe sex using contraceptives is for monogamous partnerships only, folk who practice polyamory generally know what they're doing anyway but obviously need to be much more careful about such things. Even more or less safe sex still carries risk on infection and people who might practice safe vaginal intercourse can still catch a variety of infections from unprotected oral sex (true for both partners). Gonorrhoea or Syphilis of the throat isn't a laughing matter, it's more common than you might think and it can be both caught and transmitted by unprotected oral sex.
If you live in a major city in the western world then there are probably clinics that can do everything you want in this regard, otherwise your regular family doctor ought to be able to help you. The clinic is probably preferable since they will be much more experienced, will look for and ask questions that will not occur to the family doctor that handles casual tests now and then and are almost certainly never going to be anything but purely objective (when visiting your family GP, old Dr Brown who treated you for Chicken-pox as a child might not think to ask about whether you have experienced oral-faecal contact in the last 12 months, by rimming for example, and would thus need a Hep-A test). Clinics deal with everyone including sex workers, the staff are impossible to shock and have seen every sexual fetish, act and combination before.
What to expect from a clinic will obviously vary immensely depending on where in the world you're located. I imagine that most things are largely uniform between first-world countries but some details like dispensing medicines on-site might be depend on health-insurance or local regulations in some countries. Depending on the clinic, you may or may not need an appointment and you may or may not need to see a triage nurse first. If you have to do triage, they'll just ask you a few questions to decide how urgent it is and either send you into a doctor (Page Dr Smith, got some girl whose clitoris is about to fall off) or tell you to wait or come back when it's less busy. Once you're in, the fun begins.
Tests that are commonly done (not necessarily exhaustive):
It isn't all about the tests and they won't do all those tests anyway. Trichomonas, Thrush, Vaginosis and Pap Smears are only for females and not everyone's sexual history justifies all those tests. There are numerous conditions and infections that are looked for by visual or manual inspection rather than blood tests etc, these include but are not limited to testicular cancer, crabs, fungal infections (tinea) and all manner of sores and lesions.
Before any tests get done, the nurse is going to ask you a lot of questions. Some will be fairly generic (do you have any rashes at the moment, what about unfamiliar bumps?), some will be understandable (how many sexual partners have you had in the last three months, how many male? how many female? Have you ever been sexually assaulted?) and some will be personal (how would you rate your condom usage during anal sex in the last twelve months, Always? Partial? Never?).
Then a physical examination of your genitals, pubic hair (if any), throat and anus is probably going to occur. You lie on a couch and a nurse dons some gloves and manipulates your genitals looking for external signs of infections and general medical health (I have a birth mark that got some attention and a doctor was called to make sure it wasn't anything worrying), you roll over and hug your knees and the nurse shines a torch to examine your back passage. Depending on what you're being tested for (which depends on the answers to the preceding questions, be honest), there will be swabbing of your throat, anus, urethra and vagina. With different swabs obviously. If you're concerned about someone inserting a swab into your urethra...don't be, it's on the end of a very small probe that you barely feel, it's mildly uncomfortable but not painful. If you really wish to avoid it, ensure you have a full-ish bladder and do not pass urine for at least an hour prior to the inspection. A urine sample will generally be sufficient as an alternative to a urethral swab. I found the throat swab slightly uncomfortable because I gagged slightly on the swab but it only took a few seconds. Given the size of a swab compared to say, a penis, and the anal and vaginal swabs are obviously trivial.
After that, a doctor will take a blood sample and if the visual inspection identified anything treatable then you will probably be prescribed something appropriate and possibly dispensed it on-site. Then you're done till your test results get back, how that's dealt with will vary but in general, they're unlikely to be given over the phone and you'll be asked to return to collect them and get any necessary counselling, prescriptions or treatment.
An STI test isn't anything to worry about, it takes a little under an hour in most cases, maybe less. There's no discomfort to speak of and your staff are probably going to be very professional. It's beyond the scope of this writeup to talk about what to do with the results but the staff at your clinic are going to be well qualified to discuss your options in that regard.
If you are monogamous then you (and your partner) should get tested at least once, preferably before ending the use of condoms. If you have an active sex-life that is not monogamous, then get tested regularly, every three months might be appropriate. It is an easy and painless thing to do but it can prevent a lot of minor infections becoming serious and give you peace of mind.