Second vote for a Nilfisk here, if only because they are much better value for money (IMO) than Karcher. £65 should get you the excellent Nilfisk C110, a little bit more for the C130 which is a bit more powerful. I have the C110 and it doesn't struggle with anything I've used it for (car washing, patio cleaning, deck cleaning , etc).
I use Autobrite Magifoam and I disagree that snow foam is a gimmick. Properly mixed it sits on the paintwork for around 30 minutes, giving you time to clean the wheels thoroughly before starting on the paintwork. Because it dwells for so long it really softens up the surface muck and dirt making it extremely easy to rinse off. In order to minimise risk of damage to your paintwork you want to limit the amount of contact washing that's needed and this is where snow foams excel. Because they lift so much dirt once you rinse the car down and get on with the contact wash there's very little actual dirt that needs to be wiped off.
Wheel cleaner: I've just started using Sonax full effect wheel cleaner and I'm VERY impressed with it so far. Even without a contact wash it does a really good job of getting the wheels clean. Combined with a contact wash it's brilliant. I bought 5L of it for £55, which is pretty decent value
Tyre shine: I use (and would recommend) Chemical Guys VRP. I like it because it doesn't leave a really shiny finish. It's also great on all plastics & rubbers, both interior and exterior. I use it inside and on the engine bay plastics.
Shampoo: I use Chemical Guys Mr. Pink. It smells nice and doesn't have any other additives like waxes etc. It foams up really well and you can even use it as a snow foam, though it's not as good a a dedicated product. There are many good shampoos out there though. I'll probably try something different next time, but a ~3.5L tub has lasted me nearly 1 year.
Wash mitt: I've just bought the Dodo Juice Yeti's fist. It's very good, holds an insane amount of water but takes FOREVER to dry out afterwards (like 3 days!). I'm not sure I could recommend it as it's so expensive. I'd suggest the Meguiars Microfibre mitt. Avoid the Meguiars Lambs wool mitt...it's good, while it lasts. Mine began to fall apart about about 4 washes.
Wheel brushes: Buy something dedicated for the job. I use Wheel Woolies which many recommend but they are expensive. Some say they don't last but I've had my set for 2 years and they're still going strong.
Polish: Assuming you'll be hand polishing then I would recommend Autoglym Super Resin Polish (SRP). For the price it's very good. Cover it with either a decent wax or sealant (I use Autoglym Extra Gloss Protection (EGP), but my car is white...dark colours suit waxes better). You shouldn't need to re-wax or re-seal for 6-12 months, depending on how much you use the car.
Quick detailer: The go-to product for many is Sonax Xtreme Brilliant Shine. I buy it 5L at a time for ~£35. I use it as a drying aid (sprayed on to a damp microfibre) to avoid streaks/water spots. You can also spray it directly on the car as a "top up" later of protection over whatever wax/sealant you are using.