Most nurses know how to put you at ease. I like to close my eyes and relax when they start to prep my arm for the injection and ask them to just tell me when it's done. Normally I don't even feel it. Before my surgery they started with a pill several minutes before I went into the operating room, which started the process (I don't recall specifically what it was). It helped that my surgery was at around 6:00am and I was already kind of groggy. By the time they stuck the IV in me I was comfortably lying on the operating table half asleep and - as you've probably read many times here - next thing I knew I woke up and the whole thing was over.
For me I was most creeped out by the idea of someone cutting me around my nipples - that was the image that made me shudder leading up to the operation. But, once it was over, I didn't really think much about it. The places where the incisions were made were all numb and I was covered in the vest and gauze and padding, so these ideas didn't bother me.
Don't worry about being put to sleep. You'll - presumably - be under the care of a certified anesthesiologist who will monitor your vitals throughout the surgery.
Pain afterwards depends. I had lipo on my midsection at the same time as my gyne surgery, and experienced a good about of soreness and discomfort down there. I barely felt any pain around my chest itself. I did have drains installed for my chest, which could be a bit uncomfortable. My drains were inserted through incisions under my arms, and were mostly just awkward and felt a bit weird. Once or twice a day I had to empty the drains, which involved "milking" them (you'll learn all about this - assuming you get drains), and this could sting a little bit. Otherwise, not a ton of pain.
They'll probably give you painkillers and Tylenol. Again, my operation was more involved, but I've read on here that some people just getting gyne surgery don't even need the stronger narcotics. It depends on the person and the extent of their procedure.
Sleeping post-op was the part I hated the most. Not because of pain, but because it was annoying to sleep elevated. I had to sleep almost upright (70-90 degrees) for about a week, which sucked. For the first few days the pain meds did a good job of knocking me out, but even then I'd only sleep for a few hours at a time. Make sure you find out ahead of time how your doc wants you to sleep so you can prepare - you don't want to be constantly futzing around with pillows after surgery. Some people sleep in a recliner. I was recovering in a hotel room, so I did some research and ordered a "Zero Gravity Adjustable Wedge System" to prop me up. It definitely helped, although I paid about $150 for it (my plan was to sell it on ebay afterwards, but one of my drains leaked and left a huge blood stain. Oh well
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Anyway, hope that helps. I'm about 3 weeks post-op and all of those little worries, pains and annoyances are already a distant memory. Good luck!