Hi,
Right then, i'll try and start at the beginning. DO NOT rinse the car with just water and a sponge before you wash with the shampoo, the whole purpose of the shampoo is to lift the dirt away from the paint surface, this along with a quality wash mit or sponge will not only clean your car but will do it in a safe way. The pre-rinse you were doing was actually causing swirls and maybe even the odd scratch, i'm sure it was done with good intentions though.
You then need a decent drying towel to dry properly.
With regards to the paintwork already having swirls and scratches on, this will take an awful lot of hard work to rectify by hand, but can be done. You will need a good polish along with a hand polishing pad and you will probably only do a few panels at a time as it really will be a lot of work, but well worth it.
Right, now i've explained that I will try and recommend a few products. Oh one other thing, have you ever heard of detailing clay? That can also make a huge difference.
Shampoo - Dodo Juice Born to be Mild, Chemical Guy's Glossworks, ValetPro Poseidon's are all excellent - £5.95 - £16.95
Wash Mitt - Dodo Juice Tribble or Meguiar's Lambswool Wash Mitt (both similar products) - £8.95-£9.95
Drying Towel - Waxamomo Drying Towel - £10.95
Polish - I would suggest getting both Meguiar's 105 and 205 8oz - £9.95 each
I have suggested 2 as the 105 is quite a harsh polish that will get rid of the swirls and maybe even a light scratch but may leave marks of it's own, you will then go over with the 205 to get a fantastic deep shine.
Polish - Lake Country Cutting, Polishing and maybe a Finshing Pad
http://www.waxamomo.co.uk/productcart/pc/viewCategories.asp?idCategory=75Wax - Depends on budget really, Meg's 16 is £11.95, Collinite 476 £14.95, Pete's 53 £22.95, Dodo Juice circa £30.
And or sealant - Jetseal 109 - £22.95
You will then need wax applicators, a good 4 or 5 quality microfibres towels.
That should sort the car out, and then you could add the clay I mentioned, wheel cleaners, glass cleaners etc etc
Have a read and let me know what you think.
Chris