Snarkypup, it helps to understand this if you can somehow lift the rear wheel up while looking at the derailer, like on a bike stand, or just having someone like your boyfriend lift up the seat and let you pedal. Then shift the derailer on back to see how the derailer moves. On an indexed bike, what happens is the derailer "jumps" from one discrete, set position to another. That position is determined by the derailer's angles and the cable length--NOT the position of the chain. And again, the key is that the derailer's position location over each cog is a discrete spot (these are the "detents" mentioned above). In fact, the position of the chain is determined by the position of the derailer. This should make more sense if you look at a derailer closeup and from behind as you shift the gears. You could also compare that to shifting a friction derailer (there should be a dial on the index lever that, when loosened, will allow you to switch from friction to indexed and back), and note that with a friction shift, the derailer doesn't have set positions--it simply moves inboard and outboard relative to the wheel. Where it stops is determined by where you leave the lever and the cable is pulled to, not by a setting in the derailer. In both cases, the chain rests wherever the derailer lines up. This is why you have to "trim" the derailer in friction mode, because the initial shift can leave the derailer over a spot between the cogs, or slightly off-center of the cog, and then you will hear the chain scraping against the derailer and the inside or outside of the teeth of the cog. Hopefully, that will help you visualize this better.

As far as which is more useful, it's up to your preference. Nowadays, everyone is used to using indexed shifting in the same way most people are used to automatic transmissions in cars. In a sense, you don't have to think about what you're doing so much. However, indexed shifting requires a bit more maintenance, because if the derailer goes a bit out of whack, or the cables get stretched, the system can become misaligned, and then every gear makes noise or grinds. In such a case, when you change back to friction, you can use the shifters and derailers fine. I would say if you really want to learn to shift and how the bike works as you ride it, you should learn to ride comfortably and shift easily with your derailers in friction mode. Once you're very comfortable doing this, then going to indexed shifting will provide you with a possibly more convenient riding experience--or you'll feel like you don't have any more interest in giving up that control over the bike. But it's up to you.

Because of the difficulty of cross-chaining, you never want to have that big-big combination: don't put the chain on the big ring in front with the big ring in back.

Best in such a situation as you describe is (a) as you're going downhill, make sure you've upshifted the REAR a few cogs first, and THEN shift the front from the little to the big. This way, you're on a middle gear. If you still want to go faster at that point, then you shift the rear to even smaller cogs (outside). But as you are about to go uphill, again, you want to get to the middle rear gears first, and then shift the front. In both cases, you're trying to avoid the "cross chaining" others mentioned before.

One thing to remember in general is, if you're heading downhill (or are on flat land), then shifting to the bigger cog in front should be fairly easy; the thing is, the more teeth you're moving the chain ONTO, the slower the shift will be, and the more it might "make noise." There will be a bit of a delay before this engages, also, because of the extra teeth. And as noted before, you want your legs to be applying lighter force to the pedals as you shift; this will make either derailer perform better and with fewer issues.