All is takes is some basic art supplies (I recommend watercolors) and your own curiousity. No need to worry if you are doing it "right". You will start to let go of the strangling pressure to be perfect and will feel your mind opening to creative possibilities and spontaneous beauty.
Gently remove a page from the book and use whatever art supplies bring you joy (and are compatible with paper) to add color and details. Play and see what feels fun to you. Follow your bliss. Pick a color that appeals to you and go for it.
If you're not going to take a risk on a coloring book, when will you?
Keep in mind that an abundance of wet media can soak the page, leading it to warp or tear. Don't worry about it -- just use water in moderation. Investigate and see what happens.
What feels right to you is right for you.
If you decide to sign your finished artwork (a matter of personal preference which is entirely up to you), enjoy letting your signature be an expression of who you are. It might be minimal. It might be flamboyant.
I encourage you to hang your completed artwork where you can see it in your day to day life. Let the sense of delight that you put into making it live on in the work and become an everyday part of your life -- and a part of the lives of everyone who comes into contact with you.