Correct gear names?
 

I had riden the same roadbike for 20 years which was a 12 speed. Thus when in the small front chainring I knew I was in gears 1-6 and with the large in gears 7-12. I now have a new bike, 24 speed, with three front chainrings. Seems confusing to try and think of gears 1-8, 9-16 and 17-24. Is this the correct way to think of the gears or are they referenced ie...3 / 8 for gear 24?

When discussing with others, those numbers are beyond meaningless. First, they imply a particular order, which if you work out the ratios, isn't the same as any reasonable numbering scheme you could come up with and actually remember. Second, is that other bikes may have larger or smaller gears in those same positions.

It's only slightly more helpful to refer to which ring, (small, middle, big), then which cog. Even so, my bike's gearing could be different than yours and it wouldn't have much meaning to me.

The number of teeth on the gear gives me something to compare to.

For instance, I have three different cassettes (the rear gear cluster, whose individual gears are also known as cogs) for one of my bikes. If I say I climbed the big freaking hill in my 38/21 it could mean second, third or fifth, depending on which cassette I had mounted. But since I've given the number of teeth, it doesn't matter what the position is on the cassette.

One could do the math and find a similar ratio in a completely different gear combination on your bike. But knowing the ratio, you'd know how hard I'd have to pedal. The numbers don't give that information.

See also: http://www.sheldonbrown.com/gears/

FWIW, your description tells me you've gone from a six-speed double to an eight-speed triple. That's the way things are referred to in general these days. How many in back, and how many in front.

The usual way of measuring gears is with "gear inches".
No-one uses 1-8,9-16 etc.
When considering your progression from one gear to another, esp over the overlap between chainring, it doesnt matter much which combinations you use. I usually use a different progression going up the gears than coming down for no particular reason.
The only combinations to remember are the crossover ones which can lead to excessive chain wear. They work and you sometimes find that you have been riding in a crossover gear but they are best avoided.

As with all communications, think about your audience and your goal. If you're actually concerned with the leverage, or you're comparing bikes, or you're trying to describe the gear ratios to someone else, then gear inches makes sense. But if you're just trying to remember which gears you used the last time you went up that hill, then something like 2/2 or 1/3 (middle front, second largest rear, or smallest front, third largest rear) strikes me as being easier.

Don't over complicate.

Think of your bike as having 3 gear ranges corresponding to the front chainrings. One for relatively flat surfaces, one for uphills, and one for downhills or the two times per year that you get a tailwind.

Within those three ranges, use the rear derailleur to fine tune your gearing. If you think that it's too hard to pedal, shift into an easier gear. If you think that your feet are spinning too fast, shift into a harder gear.

I have an 8 speed with a double, and not only do I not name the gear I'm in at the time, I don't even look to see what one it is. I just go along and upshift when it seems like I need to and downshift when I think I need to. Once I feel I'm really going fast, I'll move over to the large ring and if I need to stop or slow way down, I go over to the small ring. I'll know what chanring I'm in at any point, but not what cog. Maybe I'm odd, maybe not. I think that unless you're racing (or just plain OB-CON) you don't really care, right?