I'd first say I pay full attention to the wheels every time I wash the car, even if I'm not doing much with the rest of the car. The wheels basically will always be dirty, no matter what.
I've been trying out Ammo NYC's stuff so far. Just used them once, but I like them! (Note, like yourself, I'm new to this heightened interest in keeping the car clean, so I've not used a ton of stuff to compare.) Plum wheel cleaner to spray on and let sit for a bit while I attend to other places.
I use a stiffer plastic tire brush for the tires and a plastic longer brush to get inside and around the rims and wheel wells. Boarshair/hogshair is good, but my plastic one was the one at the local store when I needed something.
I'm looking to get that one from Adam's at some point...
Any brush really should be fine as long as you're not actually rubbing really hard. The soap should do a huge amount of the work for you. Really you're just wanting to agitate and move the gunk around; once freed, the soap will carry it away.
I use a microfiber mitt that I use only for wheels, but I otherwise wash with my normal car soap (this is after letting the Plum wheel cleaner sit, so it does a bunch of the work). Spray/rinse with rest of the car. Some of this I'd change if I wasn't tying up a coin-op bay while doing this extra work, but it works for me so far.
When drying, I use one microfiber towel per wheel.
And afterwords, Mud Tire Gel for my tire dressing. It's not super shiny like some people like, but does look nice anyway.