How do I start juicing?
I recommend this juicer I have it and use it every week to make lemonade:

Peel 6-8 lemons

Add to juicer

Mixing in a large pitcher, add water and sugar to taste.

Refrigerate


It's absolutely delicious. It's all I drink, no more soda.

Last week I made cherry juice. De-pitted the cherries and repeted the same steps as above.

I've also made carrot and beet juice, but they test very strong and work better when you mix them with other juices - orange juice for example.

My experience with juicing grapefruit is not good. The juice comes out very bitter and tastes very strong. 

In general, to reduce the bitterness in citrus juices, peel the fruit first before adding it to the juicer. As in the lemonade example, I sometimes use the lemon peels to make Limoncello, or keep a few in the fridge to garnish my coffee. If you have no use for them, they will at least make your garbage smell very good.



Maintenance 

You need to clean the juicer after each use, which can be discouraging. If you have someone who lives with you, have them help. My husband peels and juices the lemons, I wash the juicer parts while he mixes the lemonade in the pitcher. It takes us about 15 minutes from start to finish.



Experiment

You can throw anything in the juicer as long as you remove large pits and cut it to a size that fits the opening. It will fit a whole lemon but you need to halve an apple. 
The juicer  - at least the model I have - also comes with a recipe book. Not that I ever used it. I just use whatever I have left over and try to make something tasty.