Washing up liquid is a definite no-no as it contains salt and will remove the wax from the car.
A car shampoo is better, anything sort other than washing up liquid will do, even Asda smart price.
Detailing is as complicated as you make it, some people spend a fortune on specialist products, I prefer to spend a moderate amount and keep it simple.
Turtle wax original is well proven, but I am sure there are newer formulations available that are easier to use and which last longer.
I am currently using Zip wax shampoo, Turtle wax platinum precision car wax, and Turtle wax platinum extra gloss sealer over the top. I also use Colour magic on occasion to help mask the small scratches and swirls.
I get a great shine and without too much effort. I have tried Autoglym, Simoniz and Car plan Triplewax polishes, but none really come near to the Turtle wax for ease of use shine and value IMO.
Others will have their own preferences with what they use, like they do with oil and tyres.
Giulietta 2.0JTDm (170), BMW 335i Cabrio, BMW 120d