Step by step
How to make wudu
Wudu is washing parts of the body with water to be clean before prayer. Allah told us to do wudu because He loves cleanliness. Washing each part once is enough; washing three times is the Sunnah (the Prophet's way). Keep this order, and do the steps one after the other without a long break.
Step 1: Intention (Niyyah)
Make the intention in your heart that you are doing wudu for Allah. The intention is in the heart, you do not need to say it out loud.
Say
بِسْمِ اللَّه
Bismillah
In the name of Allah
Step 2: Washing the Hands
Pour water over both hands and wash them together three times, up to the wrists. Make sure water reaches between your fingers.
Step 3: Rinsing the Mouth
Take water with your right hand and put it in your mouth. Swirl it around inside your mouth, then spit it out. Do this three times.
Step 4: Sniffing Water into the Nose
Take water with your right hand and gently sniff a little into your nose. Then blow it out using your left hand. Do this three times.
Step 5: Washing the Face
Wash your face three times. Your face goes from the top of your forehead (where your hair starts) to the bottom of your chin, and from ear to ear.
Step 6: Washing the Arms
Wash your right arm three times from the fingertips to just past the elbow. Then wash your left arm the same way.
Step 7: Wiping the Head
Wet your hands with water. Wipe over your head once from the front to the back, then bring your hands back to the front.
Step 8: Wiping the Ears
Wipe the inside of your ears with your index fingers and the outside with your thumbs. Do this once.
Step 9: Washing the Feet
Wash your right foot three times up to and including the ankles. Make sure water goes between your toes. Then wash your left foot the same way.
Step 10: Dua After Wudu
After you finish wudu, say this dua (prayer).
Say
أَشْهَدُ أَنْ لَا إِلٰهَ إِلَّا اللَّهُ وَحْدَهُ لَا شَرِيكَ لَهُ، وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ
Ash-hadu an la ilaha illallahu wahdahu la shareeka lah, wa ash-hadu anna Muhammadan abduhu wa rasuluh.
I testify that there is no god but Allah alone, with no partner, and I testify that Muhammad is His servant and messenger.
Sources
Allah commands wudu in Surah Al-Ma'idah (5:6). These steps follow the descriptions of the Prophet's wudu ﷺ in the hadith, such as Uthman ibn Affan's in Sahih al-Bukhari 159 and Sahih Muslim 226. The dua after wudu is in Sahih Muslim 234. The schools of law differ on some details, so ask a scholar you trust.