The range of gear ratios (i.e., the difference between the highest and lowest gears) determines the minimum and maximum speeds you can travel while pedaling at a reasonable cadence. A large range, such as found on a mountain bike with a triple crankset, will let you have gears suitable for climbing steep hills as well as ones suitable for high-speed downhills.

The number of speeds (or number of gears) determines how smoothly you can transition between different speeds -- if you had some hypothetical two-speed setup with a mountain-bike-like range, for example, your problem would be that the bike would be great at climbing a steep hill or speeding down a steep hill, but terrible at everything in between. In contrast, a 21-speed mountain bike (3x front * 7x rear gears) let you have a good gear for steep uphill, slightly-less-steep uphill, moderate uphill, gentle uphill, flat, slight downhill, etc.

Bikes have been getting more and more gears over the past few decades (they're up to 11 rear gears now, which combined with a triple crank would be 33 speeds). This IMO is mostly marketing-driven (Shimano wants you to "upgrade") but I suppose also has some benefits for racers who really care a lot about keeping the variation in their cadence to a minimum.

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For touring, I think 2x7 or 3x7 gearing is more than sufficient.

As far as wheels go, I think the rear wheel you already have is just fine and you should just get a front wheel of similar age and component style, as long as it's in good condition. It's your ridiculous handlbar/shifter combination that you need to worry about!