One of the most important items to have is a turntable.Mine is an Ohuhu brand that is perfect for getting a smooth , even surface. Buy an icing spatula if you don't have one. It can even be bought at your local dollar store. Again this guarantees an even surface. You can also use a regular butter knife for this too if you want.
Another must have is an icing comb, This is a set from Cake Boss that I got from Target. It creates those elegant lines that bakery cakes have . These two create two entirely different designs with wavy squiggles or elegant lines. .
Every home bakers also needs this
This is one of my favorites. The Wilton Icer which is a dream to use. It can be used to create lush roses or delicate lines of leaves. It also makes lovely puffs of frosting on cupcakes, as well as filling them with cream. There are many attachments to create all sorts of buds and blossoms. Unfortunately it doesn't come with an icing nail. I had to buy those separately through Amazon.
As for the icing. I prefer a cream cheese frosting This one is from Allrecipes and it is a classic. I have used it not only on carrot cupcakes but also on chocolate ones as well.