I don't like taking real sleeping pills (e.g., the light stuff you can buy in the drugstore or prescription meds), but I do take melatonin. I find that melatonin (which our bodies produce naturally) doesn't knock me out, but just seems to accelerate the natural nighttime sleepiness one normally feels as the evening progresses.

I don't bother trying to reset my clock with melatonin before I fly. I just take it while I fly, and then I take it for a few nights when I arrive at my destination when I go to bed, because I find that it helps me stay asleep (or fall back asleep quickly if I do wake up). And when I travel, if I take it on an overnight flight and have to change plans part way through the journey, I wake up easily and don't feel groggy during the transit period.

If you want to try it, you can buy it in the drugstore in the U.S. (not sure where you are) - usually in the "natural" vitamin section. Take it one night at home before your trip, so you know how you feel (don't experiment on the plane). There is a small percentage of people who end up feeling woozy or who have wild dreams.

My usual strategy on a long daytime flight is to try to get some sleep but not to try to pretend that I'm operating on my destination's time zone. So if I were flying 12 hours non-stop to Asia, I'd probably try to get at least 4 hours sleep (all at once, or in the form of a nap or two). I focus more on trying to get a few good nights' sleep on arrival. If I didn't feel tired but wanted to get some rest, I would take two melatonin about an hour before I wanted to fall asleep. For example, you might take the tablets when they start the dinner service. (I take them when I board the flight on overnight flights to Europe from North America.) Then I eat a meal and read or watch a movie, and generally find that I slide into sleep after an hour or so. Use earplugs and/or sleep mask if you need to block out light and sound in order to sleep.

On arrival, if I'm tired, I will take a nap almost immediately after I arrive. Some people don't do this, preferring to stay up for the whole day and then go to bed at a moderately early hour. This approach gives me migraines, so you have to see what works for you. I'll usually take a melatonin and nap for anywhere between 1 and 3 hours - more than that and I do get groggy and have difficulty adjusting to the time zone. Then I get up, go for a walk or go to the gym, and then try to stay up until 9-10 pm, and take another melatonin before I go to bed.