I just take a microfiber rag and wash it like the rest of the car with the same soap. That also allows me to get to most of the caliper. I just put the rag on the end of a stick to get the areas my hands won't reach, and then the entire wheel can be cleaned easily and quickly. It's extremely easy.
The soap I use is Mother's Gold Class "car shampoo" or however it's branded. I use it because it actually does what soap is supposed to do, and nothing else. You can get better, but not much better.
Used bi-weekly, or even monthly as I do on a car with black steel wheels will handle things just fine and keep it all looking brand new. Waxing them can also help in places the finish is conducive to such treatment.
Recovering wheels that are already filthy to make them easy to keep clean isn't particularly complicated, but it's time-consuming. It's easier off the car, but it can be done on the vehicle if you're patient and careful. Tape off and bag the tire as much as patience allows before doing this. First wash the wheel with a rag and normal car wash soap as mentioned before, but really go to town; keeping the rag clean between scrubbing events. That stuff is abrasive, after all.
Then, if it's still dirty in places you don't want it to be, simply mix up some dish washing fluid at a slightly lower concentration than you'd use for washing dishes, and apply it to all wheel surfaces, with the wheel resting face up. On the car, you will have to re-apply frequently and do a better job masking the tire.
Let it sit, but keep it wet. This stuff is incredibly powerful when allowed time; much more than most would believe. When it starts lifting off grime when sprayed, Grab that rag and scrub again, keeping it clean.
If the wheel is still dirty, re-apply in a slightly higher concentration and grab a soft toothbrush (electrics are as helpful here as in your mouth if you can spare one) and go to town. Keep it clean, keep it VERY gentle, but get everything you can. Imagine you're bushing around your gum line. Repeat this in diminishing areas until it's got everything it's going to get. Then, wash it again normally. It should be clean enough to maintain now.
On painted surfaces, a clay bar can be used to further ensure the surface is as slick as possible. The slicker it is, the less crap it will accumulate and the easier it will be to clean in the future.
The whole point is to minimize brushing as much as possible at lower concentrations of degreaser (the dish-washing liquid). You want to be as gentle as you can, as the finish has already been abused. That means starting at a lower concentration, and if that requires too much scrubbing at any point, upping the concentration slightly. This is easy, as you can just put more in the bucket. The whole process isn't that much of a pain.
Cleaning the inside of wheels is best done off the car if they're already dirty; even when nearly new. Get them as clean as possible (you can be more aggressive here if the finish isn't as good as on the face). Then, for easy maintenance in the future, clear the shit out of that barrel and wax it. You can't fuck up that part on most wheels, as the finish isn't as good as on the face. Be careful if it is, though. Sealed, they become very easy to maintain. Even if you don't end up doing the face (unsure if it's okay, and unwilling to pay a pro; it happens), doing the inside will allow you to easily clean the part few bother with because of the normal difficulty.
If you must go further to meet your desired standard, that's when specialized cleaners come out. It is important to investigate the condition of clear coat on machined surfaces after this process; especially on diamond-turned wheels. Those are incredibly difficult to refinish. Make damned sure what you're using is compatible with your finish, and check out reviews for people saying anything was stripped on their wheels, to see if it might happen to yours. Wax what you can, clear anything required (do not touch those processes until you know they are compatible with the finish), and begin normal, easy maintenance.
Now, if you're cleaning up chrome wheels, it's not actually too much different. Clean them in the same manner, and then use lots of patience and Brasso/Neverdull. The level of corrosion a few hours will remove is staggering. Break it up over several days and just leave it on the coffee table, as it's going to be quite clean if you've done what you're supposed to do. Polish whenever you're watching TV, YouTube, or whatever, and it'll be done in a week or so.
"Mag" wheels, bare aluminum, and so on are similar. Both can be waxed or otherwise coated to protect them and enable more easy maintenance, but keep an eye on what you're using to make sure it's compatible.
Keeping tires clean is good for them, as well; and keeps them attractive as they were new. To keep them new, wash them with the same car wash soap, with a brush and then a rag. If left to do it's job, it will do it admirably. Do not use tire dressing; instead use fluid designed to replenish the oils in the rubber itself, and use it infrequently. You want hand lotion; not body paint. Your tires should be a dull satin that doesn't leave grease on your fingers when touched; not covered in snot.
If your tires are already dirty, it's fine to start with "Bleche-White" and a more vigorous scrubbing to catch up on keeping them in good health. One way or another they must be cleaned first, after all.
