All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
Whoever wants to perform a prayer whether the present one or making up for a missed one, does not utter the intention and it is not permissible to do so. But it is an obligation on him to have this intention in his heart, that he wishes to pray such or such prayer.
As regards uttering the intention before the prayer, it is an innovation. The intention is what is in the heart and not what is uttered by the tongue. The Prophet said: "Indeed actions are judged according to their intentions." [Al-Bukhari and Muslim]
The same thing applies to the prayer of 'Eed Al-Fitr and 'Eed Al-Adh-Haa and other prayers; the intention is not uttered. But if one wants to pray a given prayer, and had the intention to perform that particular prayer, then this is enough. So one just says 'Allaahu Akbar' and does not say before it: "I have the intention of praying such and such prayer", and the like.
We draw your attention that it is not permissible for a Muslim to delay the prayer from its prescribed fixed time without a sound and valid Islamic excuse. But if one delays the prayer from its fixed time, he has to make it up immediately and it is not permissible for him to delay it until its time comes again the next day. It is an obligation on him to perform it immediately and repent to Allaah.
You have also to be careful about your friends who could spoil your religion and draw you to what causes the anger of Allaah and what would make you unhappy in this life and the Hereafter.
Finally, for more benefit, please refer to Fatwa 86405 about the ruling of residing in a non-Muslim country.
Allaah Knows best.