Here are very basic answers:
1. A tank cycles when waste decomposes creating ammonia which turns to nitrite then nitrate.
2. Corals grow by photosynthesis. They are not plants though. Confused? They are animals, but they have a symbiotic relationship with a microscopic algae. More info to answer your questions.
3. How big the tank depends on how big your wallet it
30 is a great starter since there are so many options. The bigger the water volume, the less concentrated your mistakes will be.
Important Information for Those New to the Aquarium Hobby - Aquarium Advice
Cycle your salt tank - Aquarium Advice