+1 on 2 buckets wash method, then claybar and polish. Don’t forget to seal it.
Waxing car is so yesterday.
For more info see http://www.detailingworld.com tonnes of know how.
Shining up your car isn’t a hard work, its more like developing relationship with your pride and beauty. It is possible to polish it up by hand, it won’t be as easy as if it was done by machine but its doable.
You need to understand your paint and carefully pick your car products, then you may join sunday polishers league 8)
p.s. waxing is actually good, but for car shows (carnauba wax is king) only, as its not really that durable.
Paint correction and sealants thats where its at. But by the sound of it, your car should be plenty good enough with normal tidying up.