To decode the specs:
26x2 is the tyre diameter and width (26 inches by 2 inches). These fit onto 559mm rims, which are the mountain bike standard and you'll find the size marked in the ETRTO sizing on the tyre too in the format ??-559 (This is useful to know because there are about four different sizes known as 26inch, which are all incompatible with each other)
60tpi refers to the fabric carcass of the tyre and stands for 60 threads per inch. The more threads per inch, the harder it is for a foreign object to penetrate. This is a good high count meaning your current tyres have decent puncture protection.
Wire bead - the bead of the tyre is a steel wire. More expensive tyres have a kevlar bead which allows them to be folded for easier packaging/storage and to be lighter.
With the wheel, double wall refers to the rim. A double wall rim is stronger than a single wall.
32h refers to the number of spoke holes. 32 is a good high number meaning the wheel will be strong (36 is usually the most a wheel is built with except for the extra strong wheels on tandems or cargo bikes)
With the chainset, the 42-32-22 refers to the number of teeth on the chainrings at the front. Since you've said they're replaceable ones, you have the option to customize your drivetrain a bit.
Changing the tyres would make the bike roll a bit better on road. A slick such as
Schwalbe City Jet would be fine if you are sticking to purely road use, or a touring style treaded tyre would be good if you still want a bit of off road capability. I've tried and like
Continental TourRide which has good puncture protection and seems to work on most surfaces (except for mud).
Schwalbe Marathon is another one I'd suggest, which has better puncture protection still, but costs a bit more. There are plenty of alternative choices which others will no doubt suggest.
Wheels wise - those look like decent ones and if it was my bike I'd be inclined to stick with them.
With the chainrings, I'm not sure what you want to achieve. I suspect a little more top end gearing? If that's the case changing to a 44 tooth big ring will probably do the trick. Do you know what spec the cassette at the rear is?
Apologies for waffling on a bit. I hope this makes sense and I'm not teaching you to suck eggs.
