I dont know much about software development so please forgive my lack of knowledge on the subject.

I am looking to build a subscription based service for hair stylists that will include a website with appointment setting features. I have found some other really good appointment booking software that is already available. What is the best way to go about designing a similar software and features menu without copying other peoples work? It seems to me that a lot of the software that I have seen has similar features. I have a good prototype of what I want everything to look like as well as a list of features I want to include but am clueless on where to go from there. Is it illegal to have someone design software that is just like one that is out there?

Again sorry for the lack of knowledge on the subject I am just trying to figure out the best way to go about this.

Not illegal as long as you don't take their program and call it yours. Make your program a little different than others already on the market and you should not have any legal problems. But you will want to do a lot of market research to see if there is any market for your program. If Product X does everything anyone would want and sells well then you might have a really tough time breaking into their market.

Define the requirements , draw the guis , design it and code it , you do not have to worry much about copyright violation as long you are not actually copying it from somewhere.