The water for ablution must be kept in a covered vessel.
(Saheeh Ibn-e-Khuzaima, Sanad Saheeh)

When get up in the morning, then before performing ablution, must wash the hands thrice, before dipping the hands into the vessel containing water for ablution.
(Saheeh Muslim)

Say Bismillah before beginning the process of ablution.
(Nasai, Sanad Hassan)

Then with a handful. (Ahmad Sanad Saheeh) of water in the right palm, wash both the hands thoroughly, by rubbing up to the wrist, so that the hands are thoroughly clean; also clean the fingers of both hands by intermingling with each other. Repeat the whole process three times.
(Saheeh Muslim, Tirmizi, Sanad Hassan)

Then with a handful of water in the right palm, put little water into the mouth, and spill out after bubbling for a while in the mouth, put the remaining water into the nose, and suck in excess, if fasting, then avoid sucking in excess, then holding the nose with the left hand, blow out. If have not used Miswak then clean the mouth by rotating the finger inside. Repeat the whole process three times; then every time from the handful of water, use some water for bubbling in the mouth, and the remaining for sucking in the nose.
(Saheeh Bukhari, Saheeh Muslim)

Note: There is no evidence or proof for using handful of water, separately, for sucking in the nose and bubbling in the mouth, just one handful of water has to be used for both the acts.

Then containing a handful of water in the right hand’s palm, wash the face by joining both hands together, rub the side corners of the eyes, and wash the face in the similar way, for three times.
(Saheeh Bukhari)

Then pour little water under the chin, and comb the beard from down upward, three times.
(Hakim, Sanad Saheeh)

Then, with a handful of water, wash up to or even above the elbow, repeating the process three times, then wash the left hand up to elbow or even above, with a handful of water, three times.
(Saheeh Bukhari, Saheeh Muslim)

While washing rub both the hands thoroughly.
(Hakim, Sanad Saheeh)

Then make both the hands wet, with a handful of water in the right hand, then draw both the hands over entire head (in a gliding manner); place both the hands on the forehead, and draw back ward up to the nape of the neck, and then similarly bring back both the hands up to the forehead.
(Saheeh Bukhari)

Then make wet, both, the fore fingers and the thumbs, insert the forefingers into the ears, then glide the fore fingers inside the ears, and the thumbs outside the ears.
(Hakim, Sanad Saheeh)

Then pour water on the right foot, with the right hand, then wash with the left hand, rubbing thoroughly up to the ankles, or even above the ankles, and comb the fingers of the foot, with the index finger of the left hand, then in the similar manner wash both the right and left foot, alternately, three times.
(Saheeh Muslim)

Then sprinkle a handful of water on the gusset of the pajama.
(Rawah Ahmad, Sanad Saheeh)

Then recite the following words:

اَشْھَدُ اَنْ لَّاۤ اِلٰہَ اِلَّااللّٰہُ وَحْدَہٗ لَاشَرِیْكَ لَہٗ وَاَشْھَدُ اَنَّ مُحَمَّدًا عَبْدُہٗ وَرَسُوْلُہٗ۔

Translation: I stand as a witness that Allah alone is the absolute Supreme Lord, the absolute Remover of difficulties, and there is none who shares His absolute authority. He is the sole Almighty Allah, and witness that Muhammad SallAllahu Alaihi Wasallam, is His prophet and His man (slave).

By reciting the above Kalimah (words) all the eight gates of the paradise are opened for the reader.
(Saheeh Muslim)

Any one who recites the following supplication after ablution, then the supplication is sealed after writing it on a piece of paper, to be opened only on the Last Day, not before that.
(Nasai, Sanad Saheeh)

سُبْحٰنَکَ اللّٰھُمَّ وَبِحَمْدِکَ، اَشْھَدُ اَنْ لَّآ اِلٰہَ اِلَّآ اَنْتَ، اَسْتَغْفِرُکَ وَاَتُوْبُ اِلَیْکَ

Translation: O,  Supreme, the Almighty Allah, You the absolute holy and sacred, all praise be to You, the absolute holy and sacred, free from every defect and short coming, I stand as a witness that You alone are the absolute Omnipotent, and I seek forgiveness and repent from You.

Wipe the face with a towel, after (Wazoo) ablution.
(Tirmizi, Sanad Saheeh)