I worked in our endoscopy lab today. I promise you, the prep and the anxiety over the test is actually worse than the actual procedure itself.
I can't speak for other endoscopy centers, but I can walk you through how ours is.
If you're really worried about being 'cleaned out' enough, you can actually switch over to clear liquids a day or so before the test, then take some Dulcolax tablets or similar laxatives the day before you do your actual prescribed prep. Ask your MD if this is okay, but all of the women I work with have done this with no problems.
Start your prep the afternoon before your procedure, so you won't be up all night running to the bathroom. Nothing to eat or drink at all after midnight.
The day of your exam, your doctor may or may not tell you to take your normally prescribed medicine...of course, check with them beforehand. You'll probably have to show up at the center about two hours before hand, the nurses will check you in, go over your history and allergies again, and make sure all your paperwork is in order. They will ask you to get totally undressed, and take off your glasses and take out your dentures if you have them. They will start an IV in your hand or forearm, and sometimes start some IV fluids through them.
Once you get in the actual procedure room, they will do both procedures back to back. You will be hooked up to a heart monitor, a blood pressure monitor, and an oxygen saturation monitor. Sometimes they will go ahead and put a nasal cannula on you and give you oxygen through it during the procedure. The doctor will tell the nurse to administer the medicine through your IV. Most places use a combination of pain medicine and a sedative, this is called conscious sedation and you will be in 'twilight' sleep, and probably won't remember anything about the procedure. You will still be able to follow commands and might even be able to answer questions, but you should be comfortable. If you complain of any pain, or even give the MD non-verbal clues that you are uncomfortable, most of them will give you more medicines. Sometimes pts receive sedation from a CRNA or anesthesiologist instead of a nurse.
If the colonoscopy is first, they will ask you to lie on your side, and draw your knees up to your chest. You will be draped prior to the procedure, so don't worry about your hiney hanging out. By the time they uncover it, you won't care. The doctor will do a digital rectal exam, then pass the scope. There will be quite a bit of air introduced into your colon during the procedure, as well as some flushing and suctioning done. We encourage patients to pass gas during and after the procedure. You will be absolutely miserable if you don't!! After the MD advances the scope up to the distal end of your colon, or the cecum, they will slowly back it out, looking for abnormalities along the way.
If you have the esophagogastroduodenoscopy first, they will probably spray the back of your throat with some numbing medicine, then give you the IV sedation, then put a plastic bite block in your mouth to protect your teeth and to make passage of the scope easier. The EGD is a much quicker test than the colonoscopy...not as much 'real estate' to cover. The doctor will tell you to swallow while they are trying to pass the scope.
The doctor can take biopsies through the scope, they can remove polyps, band bleeding vessels, and cauterize bleeding as well.
Afterward, you will wake up and hopefully not remember anything after getting in the procedure room. I can't tell you how many patients wake up and ask, "When are you guys going to start?" You will feel a little groggy an sleepy the rest of the day, and will not be able to drive. You may have quite a bit of gas afterward, too, just let it out. After a couple of hours, you will be discharged home and can start drinking liquids and advance your diet as you tolerate it. Your MD will give you a preliminary report of what they saw during the exam, and if they biopsied any places or removed any polyps, they will have to send them off for pathology, and you will have a followup appointment to discuss the findings.
Hope this helped a little, and hope your tests go smoothly!
Here is my experience if you want to know:
DH drove me as you are not allwoed to drive yourself home and we stopped at Burger King so I could get food ASAP. After I ate I felt totally fine except for a very minor sore throught (lasted about 2 days) and I even asked to stop and but groceries before heading home.
I see him at least once a week, so at least in your case you won't have to worry about that. 
(by the way, I say he 'was' my friend's hubby not because we're no longer friends but because sadly she just passes away in April) 
And that was just a dentist!
