What must I do if I opened the email after swearing to fast for two months if I entered it, and what are the consequences if I opened another email after swearing to fast for two months if I entered any email of mine?

If you swore by Allah not to enter Messenger and then entered it, you have broken your oath and must pay an expiation for a broken oath (kaffarat yamin (definition)). As for fasting two months for not entering Messenger, this is an oath of obstinacy (nadhr (definition) lajaj), and its maker has the choice between fulfilling it or paying the expiation for a broken oath. If the reason for the oath was the words of friends and your fear of entering forbidden websites, and this reason has ceased due to your commitment not to enter them, then it is permissible for you to enter Messenger, and no expiation is required of you, because the context of the oath specifies it.