Yes, dish soap will dry the coat out real bad. Not to mention it's really hard to rinse out because it just keeps getting soapier and more soapier you will spend a long time getting the soap out, and if you don't get it all out you will end up with 'hot spots'. However, there is one kind of dish soap you can use without it drying the coat out like other dish soaps and that would be Ivory dish soap. When I groomed dogs, we used it on puppies to get rid of fleas because the flea shampoo is too strong for them.
As for general grooming goes: brush, brush, brush! Lol It's basically a never ending cycle. Be sure to pick the feet DAILY to prevent infections such as thrush, it's very important to keep the feet real clean (thats basically the foundation of a horse). As for the mane and tail...if you want to keep the tail real long, don't brush it out...just finger through it with your fingers to avoid pulling the hair out and use detangler. The mane can be brushed out, but you can also use your fingers for that too.
The importance of grooming: Other than keeping your horse clean, grooming is important to keep the blood circulating to the superficial tissues which is the skin. Also, during the seasonal changes (from winter to spring/summer) it's real important to get their winter coats off. Why? Because with the temperature changing from 30's to 60's and up, when you are working your horse and they still have a winter coat they can overheat real easily since the temperature is higher. There's more I can add, but I don't feel like typing, so I figured I'd just go over the most important stuff hehe