I recently took my first flight with a cat, and thought I'd report on it, in case this is useful for other first-timers.
I flew with a 17-year old cat. He flew with me in the cabin. I didn't sedate him. I put him in a Sherpa soft-sided bag (size Medium; fit under the seat in front of me nicely, with only a little bit of squooshing of the bag). I flew on United, on a 757ps airplane, in the economy cabin. I booked a window seat, to ensure that others didn't kick the pet carrier when getting out of their seat, and I was glad I did. Also, at least on that plane, there is a bit more room under the seat in front of you for middle and window seats than for aisle, so that's something you might want to take into account when booking travel with a pet in the cabin.
The trip went completely uneventfully. I was a bit concerned about the cat going nuts or meowing/crying through the whole flight, but he settled down very quickly. I was glad I was able to take a direct flight.
Going through security was also straightforward. Of course I had to take him out of the pet carrier and carry him through the metal detector. I was worried about getting him back into the pet carrier afterwards, but that turned out to not be a big deal; he normally hates the pet carrier, but in this case he seemed fairly eager to get back in the pet carrier (I guess it felt like a safer spot than the big wide-open security area). The TSA folks were supportive and helpful.
I was a bit paranoid about taking him out of the pet carrier, for fear that if heaven forbid he got away from me somehow, he might run away and I might never find him again. OK, I know, pretty unlikely, but I really don't like to take chances on that... I had initially intended to ask for a private screening, so that I could only take him out of the pet carrier in a closed room with all the doors closed. But it turns out this didn't work: at the airport where I was flying out of (JFK), their private screening area was just a corner of the screening area with some dividers to block sight lines. Oh well.
I did put a harness on him before putting him into the pet carrier. When it was time to take him out of the pet carrier to go through security, I attached a leash (a short lead) to his harness and around my wrist, as an extra safety measure in case he got all crazy-squirmy. But it wasn't an issue.
As for getting him into the pet carrier in the first place before the trip, I wasn't sure what to expect, since I know he hates going in there. The trick I used was to tip the pet carrier upright, set it back by a wall, pick up the cat from another room (where he didn't see the pet carrier) by the scruff of his neck, and walk backwards towards the pet carrier (so he couldn't see he was heading towards it), then gently lower him into the carrier. The trick was to hold his weight entirely by the scruff of his neck, without supporting his weight with my hands under his hind legs. Apparently, cats have an instinct to not fight it when they're carried by the scruff of their neck, because mother cats move kittens out of a dangerous location by picking them up by their scruff. It looks horrendous but it doesn't hurt the cat and really helps eliminate the squirming and fighting from the cat. The trick worked quite well.
I encountered no problems from the airline. They didn't check my health certificate. United blocked me from checking in online or via EasyCheckin, so I had to see an agent, but was all uneventful. I was charged a $125 fee. I didn't see that I got any extra service or value for the money, so I felt that I was being a bit raked over the coals. It felt like the airline charged me "because they could", so the unnecessary fee felt a bit annoying, but what can you do?
Overall, everything went smoothly, and I'm glad I did it. To others who are doing this for the first time, relax and don't stress out too badly: it wasn't as bad as I had feared.