Inexpensive wheels = donor bike. Find another cheap bike, but one with better wheels. Not only will alloy wheels be lighter in weight, the brakes will work much better.

As far as where to get parts, it depends on whether this is a one time project, or if you plan to do more. One time project = head to the local bike shop. Multiple projects = order on line (Niagara Cycle is a favorite around here, they are cheap, but VERY slow).

Since you already are into cars, you probably already have a dial caliper. If not, pick up a cheap one at Harbor Freight.

I rarely replace headset bearings, I usually clean them and regrease. BB bearings are almost always 1/4 inch, 11 per side, rear wheel usually 1/4 inch, 9 per side, front wheel usually 3/16 inch, ten per side.

Don't get too carried away with upgrades. It all starts with the frame (which is low end on that Berkeley). The Berkeley is basically an entry level bike, from a good company. At some point, it is a lot cheaper to just sell it and buy something better. But it will be worth doing the basic maintenance and finding some better (used) wheels.