Short answer
You thank Allah in three ways: in the heart by knowing every blessing is from Him, on the tongue by saying alhamdulillah, and with your body by using His gifts in ways He loves. Allah promises, "If you are grateful, I will surely increase you" (14:7).
Gratitude to Allah is wajib in general, and using His blessings in sin is ingratitude. Saying alhamdulillah after eating, drinking and sneezing, and on good news, is sunnah. Prostration of thanks (sajdat al-shukr) on receiving good news is mustahabb according to the relied-upon Hanafi view.
The four schools agree on this.
Explanation
In the heart. Shukr starts with noticing. Your health, your faith, your family and the next breath are gifts, not things you are owed. Allah says, "Remember Me; I will remember you. And be grateful to Me and do not deny Me" (2:152).
On the tongue. Say alhamdulillah often, and speak of Allah's favours to you without boasting. The Prophet ﷺ said Allah is pleased with a servant who eats a meal and praises Him for it, or drinks and praises Him for it. He taught Mu'adh to say after every prayer: "O Allah, help me to remember You, to thank You, and to worship You well." See Alhamdulillah.
With the body. Use each blessing in obedience: eyes, wealth, time and strength. The Prophet ﷺ prayed at night until his feet swelled. When asked why, since he was forgiven, he said, "Should I not be a grateful servant?" Many also make a single prostration of thanks when good news comes, as Abu Dawud reports he did.
Thank people too. "Whoever does not thank people does not thank Allah" (Abu Dawud). Saying jazakallah khair is part of shukr.
The opposite of shukr is to treat blessings as yours by right, or to use them against the One who gave them. Gratitude brings more: more blessing, more contentment and more closeness to Allah.
The evidence
Quran, Surah Ibrahim 14:7
وَإِذْ تَأَذَّنَ رَبُّكُمْ لَئِن شَكَرْتُمْ لَأَزِيدَنَّكُمْ ۖ وَلَئِن كَفَرْتُمْ إِنَّ عَذَابِى لَشَدِيدٌۭ ﴿٧﴾
And [remember] when your Lord proclaimed, 'If you are grateful, I will surely increase you [in favor]; but if you deny, indeed, My punishment is severe.' "
Gratitude brings increase from Allah.
Quran, Surah Al-Baqara 2:152
فَٱذْكُرُونِىٓ أَذْكُرْكُمْ وَٱشْكُرُوا۟ لِى وَلَا تَكْفُرُونِ ﴿١٥٢﴾
So remember Me; I will remember you. And be grateful to Me and do not deny Me.
Allah commands gratitude to Him.
Sahih Muslim 2734
Sahih... إن الله ليرضى عن العبد أن يأكل الأكلة فيحمده عليها ...
"Allah is pleased with a servant who eats a meal and praises Him for it."
Praise after a blessing pleases Allah.
Sahih al-Bukhari 4836
Sahih... أفلا أكون عبدا شكورا ...
"Should I not be a grateful servant?"
Gratitude is shown through worship.
Sunan Abi Dawud 1522
Grading to be checked... اللهم أعني على ذكرك وشكرك وحسن عبادتك ...
"O Allah, help me to remember You, to thank You and to worship You well."
A sunnah dua asking Allah for help to be grateful.
Where the ruling is recorded
- al-Durr al-Mukhtar with Radd al-Muhtar, Bab Sujud al-Tilawa, discussion of sajdat al-shukr. Gives the fatwa that the prostration of thanks is mustahabb, following Abu Yusuf and Muhammad.
- Ihya Ulum al-Din (al-Ghazali), Kitab al-Sabr wa al-Shukr. Explains gratitude of the heart, tongue and limbs.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.