Grape Juice is made best with Concord Grapes. They don't always taste the best themselves since they are full of seeds, but they are perfect for juicing! Making grape juice is actually extremely easy and takes a lot less work than you would think. The trick is using a good steam juicer! (I recommend either the Back to Basics Steam Juicer, which is aluminum, or the Victorio Steam Juicer, which is more durable because it's stainless steel. The Back to Basics Juicer is less expensive, but there are people who swear by the Victorio brand. It's your choice, I think both are great!)
Take each grape off the vine, for freshness make the juice the same day as pulling them off, and simply wash them off. You don't even need to worry about pulling the stems or seeds, just put them into the top of the juicer. Fill the bottom with hot water, put it on the stove at about medium temperature, and as the steam rises to the top, it will deflate each grape and pull the juices out and down through the hose. NOTE: MAKE SURE YOUR HOSE LEADS TO A SEPARATE BOWL/JUG TO COLLECT ALL THE JUICE!
Regarding taste, add however much sugar you want to make it taste the way you prefer. I've made batches without sugar and others with a lot depending on the taste of the grapes themselves.