"The apple is not small, but big."
Translation:Az alma nem kicsi, hanem nagy.
Can you say, "Az alma nem kicsi, de nagy." So, can you use 'de' instead of 'hanem'
Is nagyon nagy but in another case or something like that? I find it interesting that the two words seem related
Yes, nagyon is an adverb derived from the adjective nagy. This translates to English pretty literally: if you take "great" as the meaning of "nagy", then "nagyon" is "greatly" (although you would almost always use "very" as the translation.)
Most adjectives can undergo this transformation to an adverb, including adjectives that would sound decidedly weird as adverbs in English.
Szép - szépen (beautiful, beautifully), kellemes - kellemesen (comfortable / comfortably), lassú - lassan (slow / slowly), friss - frissen (fresh / freshly) and so on are all examples of the same thing.
Mizimano is right. The verb "to be" is not used explicitly in the phrases of present tense.