電話番号を教えてあげる。
In English we usually say "I'll give you my number". Can I just say 番号？ Would you say ケイタイ番号 for cell phone？ この表現は自然ですか？

that's perfectly correct, but a little litttle bit like a phrase in a textbook. If you would like to go casually, you can say ○○○教えてあげる(ね/よ) or ○○○教えるね/よ. 携帯番号, ケータイ番号, 携帯, ケータイ, 電話, and 電話番号 are all ok to use. を is sometimes dropped when a conversation is casual.
In formal situations like an important meeting with a client, I would use a phrase like 当方の携帯番号をお伝えします。