I personally think of it this way.
教える although meaning to teach, can be used in place of the English to tell.
For example,
- 電話番号を教えてください。Please TELL me you phone number.
- 天体物理学{てんたいぶつりがく}について教えてください。Please TEACH me about Astrophysics
In this instance please TEACH me your phone number would sound somewhat odd (similar to I never want to forget your phone number). However teaching someone about Astrophysics would not seem odd at all.
お知らせする would mean something close to, to inform / notify.
For example,
- これから明日の会議についてお知らせいたします。 From now, I will inform you about the details of the conference tomorrow.
OR
From now, I will notify you as to the details of tomorrows conference.
- 電話番号変更の知らせ。 Notification of change of phone number.
Whilst not explicitly used for this purpose, although probably able to be understood, it would seem a bit odd to substitute 教える with お知らせする in most circumstances. Although this does depend on what exactly you are saying.
- 10時に連絡先を教えます。I will tell you the contact details at 10 o'clock.
- 10時に連絡先をお知らせします。I will inform you as to the contact details at 10 o'clock.
After a bit of research, 陳べる does not seem to be in use and has been replaced by 述べる.
述べる means something close to express, state or mention
述べる implies more of giving a statement
- 意見を述べる to express / state ones opinions (seemingly the most common use of 述べる is when it has 意見 attached to it)
- 理由を述べる to give / state a reason
In this sense, 教える can't be replaced by 述べる without altering the meaning of the sentence.
- 暗号を教えてください。Please tell me the passcode.
- 暗号を述べます。To state the passcode.