If you can, wash it using the two bucket method with grit guards, one bucket with just water and one with shampoo and water. Wash the car with the shampoo/water, then rinse the mitt in the clean water bucket. Keep emptying the clean water and refilling to ensure you're rinsing all of the stuff off of your mitt and not just putting crud back on it.
I use two wash mitts, one for the lower areas where the most dirt from the road collects and one for the upper, more visible areas. Wash it from top to bottom, not bottom to top.
For the clay bar, it's best to get a good lube spray. You can use most quick detailing sprays too for this. Spray a panel, clay it, then wipe it off. You'll probably want/need to wash the car again after this. If not, you can probably use the shampoo/carwash stuff you have. Most of that stuff can be used to clay the car. You want to lube the surface with something like that, not water, otherwise you'll just do more harm than good.
If you're just washing it to get dirt and some contaminants off, you don't need to polish and wax it. Polishing is meant to remove surface swirls and paint imperfections. Waxing is just applying protection to the freshly polished surface.
I wouldn't attach anything to a drill to do this.
In your case, I would just wash it and clay it and apply a coat of wax by hand. I think you'll be fine with that. In order for you to polish and all properly, you'll need a lot more equipment than what you have on hand it seems.
'11 BMW E92 ///M3 - ZCP and DCT
'13 Toyota Tundra - 4x4 Platinum CrewMax, 5.7l iForce V8