Usually I start making my juices with fruits because they add nice water to it. So I would usually begin by adding a few oranges to get some vitamin-C in my juice. After that I would like to add the grasses and the most important part - the broccoli. Honestly, using fresh organic broccoli on a daily basis and some other cruciferous vegetables can be a life changer for most of us. Some things are better when you eat them fresh and don't use just supplements for it. When that is done, I usually add some veggies and berries to the mix depending on what I have and what I feel like. Some days it might be strawberries and carrots, and other days it might be raspberries and asparagus.
If you need to make your juices taste good, then you can try to come up with tasty combinations and rotate them. I sugges that you still keep a some sort of grass and broccoli or other cruciferous vegetable in there to keep it as healthy as possible. Using a Jack Lalanne Juicer you can be quite certain that you will get a good juice every time no matter what you add there. Remember that it is all about the smoothness of the drink - if it isn't that great tasting you can get it down if it is smooth. IF you buy a cheap juicer you will end up with small bits and pieces in there which are going to make you vomit.