Washing up liquid is for your dishes, if you want to use it on your bike or even you car then thats your choice, BUT washing up liquid does contain harsh detergents and salt both will strip and damage whatever finish you have on your bike/car, not forgetting how well it removes grease from dishes so just think how quickly its thinning and removing the grease from your bearings.
Wax can and does add protection, however nothing will protect from stone chips, well unless you add a plastic coating, like the 3M paint shield thats used on cars, but it will add weight and also dull the look somewhat. Wax is a good quality one containing a high percentage of Carnauba will protect from stuff like surface dirt, tar, bugs and general crap weather, it also helps to keep the finish looking great and disperses water quickly.
If you want a suggestion for a shampoo, try a no tears baby shampoo, this is perfect at removing dirt without stripping wax protection or grease from bearings, even if used at 1 bottle per bucket its fairly safe, but 2 capfuls in a bucket is perfect, it works perfectly as I know, as its what I use on serveral show cars I look after. Failing this try a quality car shampoo, again a capful per bucket, not a gulg. Cheap shampoos are just that cheap, detergents are coarser and also contain salts to bring the costs down.
I only use Swissvax Divine wax on my road bike, its only £1200 a throw, but as its part of my work and the bike only needs 2 coats max a year and still beads water, even though its washed after almost ever ride it makes my life easy
