Supplicating for Good Health is from the Sunnah.

It is recommended to include a request for seeking good health from Allah Azzawajal in each supplication (du'a).

Ibn 'Umar (radhi Allahu anhu) narrated, "The Messenger of Allah (sallAllahu alayhi wasallam) used to supplicate saying: Allahumma inni a'udhu bika min zawali ni'matika, wa tahawwuli 'afiyatika, wa fujaati niqmatika, wa jami'i sakhatika [O Allah! I seek refuge in you from the withholding of your favour, the decline of the good health you have given, the suddenness of your vengeance and from all forms of your wrath]."

Related by Muslim. Bulugh al-Maram Book 16, Hadith 1558 and also in Riyad us Saliheen.

Visiting the one who is suffering an illness and supplicating Allah for their recovery is also from the Sunnah.

It is reported that the Messenger of Allah (sallAllahu alayhi wasallam) used the visit the sick and pray for their recovery and in one narration, that they drank from the water which remained after he had performed ablution.

Muawiya bin Suwald (radhi Allahu anhu) narrated that "I heard Al-Bara' bin 'Azib (radhi Allahu anhu) saying, "The Prophet (sallAllahu alayhi wasallam) orders us to do seven things and prohibited us from doing seven other things." Then Al-Bara' mentioned the following, (1) To pay a visit to the sick (inquiring about his health), (2) to follow funeral processions, (3) to say to a sneezer, "May Allah be merciful to you" (if he says, "Praise be to Allah!"), (4) to return greetings, (5) to help the oppressed, (6) to accept invitations, (7) to help others to fulfill their oaths.

Sahih Al Bukhari Volume 3, Book 43, Number 625.

The Qur'an is a Cure and so it's general recitation for those in good health as well as those who are ill is very important, as Allah Azzawajjal says;

The Holy Prophet's supplications for the Sick (sallAllahu alayhi wasallam).

It was narrated Abdullah ibn Abbas (radhi Allahu anhu) that the Prophet (sallAllahu alayhi wasallam) said: "If anyone visits a sick whose time (of death) has not come, and says with him seven times: As’alullaahal-‘Azeem Rabbal-‘Arshil-‘Azeem an yashfiyak [I ask Allah, the Mighty, the Lord of the mighty Throne, to cure you], Allah will cure him from that disease."

Sunan Abi Dawud Book 20, Hadith 3100. Classed as Sahih by Sh. Al Albani (rahimullah).

And also;

Our Mother A'ishah (radhi Allahu anha) narrated that "When any person among us fell ill, Allah's Messenger (sallAllahu alayhi wasallam) used to [pass over him] with his right hand and then say: Adhi bil baas, Rabanaas. Washfi antashaaf'i. La shifaa'a, illah shifaa'uka Shifaa'an, la yughaadhiru sakaman [O Lord of the people, grant him health, heal him, for Thou art a Greet Healer. There is no healing but with Thy healing Power] one is healed and illness is removed. 

Sahih Muslim Book 26, Hadith 5432.

And also;

As-Sa'ib bin Yazid (radhi Allahu anhu) narrated that "My aunt took me to the Prophet (sallAllahu alayhi wasallam) and said, "O Allah's Apostle! This son of my sister has got a disease in his legs." So he (sallAllahu alayhi wasallam) passed his hands on my head and prayed for Allah's blessings for me; then he performed ablution and I drank from the remaining water. I stood behind him and saw the seal of Prophethood between his shoulders, and it was like the "Zir-al-Hijla"" (means the button of a small tent, but some said 'egg of a partridge.' etc.)

Sahih Al Bukhari Volume 1, Book 4, Number 189

The best times when supplications are accepted.

1. On the Night of Power [Laylatul Qadr].
2. The final part of the night [meaning the last third].
3. At the end of the obligatory Salah.
4. Between the Azhaan and the Iqaamah.
5. An hour during the night.
6. With the Azhaan for every Salah.
7. With the rainfall.
8. While marching in a military formation for the Cause of Allah.
9. An hour during Friday. The accepted view is that this hour is in the final hour before sunset, although it is also held that it may be during the Friday sermon of Jumu'ah.
10. While drinking the water of Zamzam with a sincere intention.
11. During prostration.
12. The transmitted supplication offered by the one who wakes up from sleep [for Qiyyaam ul Layl].
13. When sleeping [after having completed ablution] after having been in the state of impurity, then awaking in the night and supplicating.
14. While supplicating "There is none worthy of worship but You, Glory be to You, verily I am from amongst the wrongdoers."
15. The supplication of a group of people following the death of an individual [for the deceased in Janaazah].
16. The supplication offered after glorifying Allah and sending salutations upon the Prophet (sallallahu alayhi wasallam) in the final sitting part of the Salah.
17. While calling upon Allah by way of His Most Beautiful Names by which a supplication is always answered and a request is fulfilled.
18. The sincere and heartfelt supplication of a Muslim for his/her Muslim brother/sister.
19. On the Day of Arafah [9th Dhul Hijjah] at the Mount of Arafat.
20. Supplications during Ramadhan.
21. While in a gathering with other Muslims who have congregated to remember Allah.
22. While undergoing a hardship and supplicating "To Allah be belong and to Him we will return. O Allah, save me from my affliction and compensate me with good from it."
23. The supplication offered when the heart becomes disinclined to Allah and sincerity weakens.
24. The supplication of the oppressed one against those who are oppressing him/her.
25. The supplications of the father in favour of/or against his child.
26. The supplications of a traveler.
27. The supplications of the one who is fasting [up until Iftaar].
28. The supplication offered at the time of breaking the fast.
29. The supplication of an individual who has been compelled to something against their wishes.
30. The supplication of a just and upright leader.
31. The supplication of a reverent and pious child on behalf of his/her parents.
32. The transmitted supplication offered directly after wudhu.
33. The supplication after stoning the small Jamrah [during Hajj].
34. The supplication offered after stoning the intermediate Jamrah [during Hajj].
35. The supplication and Salah offered inside the Hijr, which is the curved precinct which joins to the Kab'ah.
36. The supplication made upon Safaa and Marwah.
37. The supplication made at Al Mash'ar ul Haraam, the consensus is that this is at Al-Muzdalifa.

Reference - Supplications & Treatment with Ruqyah from the Qur'an and the Sunnah - Dr Sa'eed bin Ali Wahf Al Qahtani. Darussalam 2009 Edition.

The 24 essential etiquettes of supplication (Dua) and the means for its acceptance.

1. Sincerity to Allah.
2. To start the supplication by praising Allah and expressing gratitude to Him, thereafter sending salutations upon the Messenger of Allah (sallallahu alayhi wasallam). The supplication should also conclude in the same manner.
3. Determination in the supplication and being certain of a response.
4. The supplication should be heartfelt.
5. One should supplicate by means of using and praising the Beautiful Names and lofty attributes of Allah.
6. The supplicant may also petition Allah by means of his/her own good deeds.
7. An insistent supplication which is offered without haste.
8. To supplicate during both times of hardship and times of ease.
9. To ask Allah alone without asking by means of others.
10. To refrain from supplicating against the family, wealth, children and yourself.
11. To lower one's voice when supplicating to a moderate tone which is neither loud (so that you can be overheard) nor too quiet.
12. To confess one's sins and ask forgiveness for them and to recognise the bounty of Allah and express gratitude for it.
13. To refrain from the use of rhyme (or poetry) when supplicating.
14. To be humble and submissive and express fear of Allah.
15. To reject injustice along with your repentance.
16. Repeating specific supplications thrice and to use the supplications of the Holy Prophet (sallallahu alayhi wasallam) which are reported as Sahih.
17. To face the Qiblah (direction of the Ka'ba) when supplicating.
18. To raise the hands when supplicating but do not look upwards at the Heavens.
19. To be in the state of ritual purity (Wudhu) prior to starting the supplication.
20. Not to exaggerate in the supplications.
21. To begin by supplicating for yourself if the supplication is on behalf of another person.
22. The supplicant's food and drink, clothing and income should be from Halaal means. They should not be involved in some type of fraud or the consumption or payment of Riba (interest) on their income for example.
23. The supplicant should not ask for the severing of family ties or ask for a sinful deed.
24. The supplicant should command good and forbid evil acts and try his/her utmost to safeguard themselves of all evil deeds and sins.

References -

Regarding not lifting the gaze to the sky -
It was narrated from Abu Hurairah (radhi Allahu anhu) that the Messenger of Allah (sallallahu alayhi wasallam) said: "People should certainly stop lifting their gaze to the sky when they supplicate during the prayer, or they will lose their eyesight."
Classed as Sahih from Sunan an Nas'ai Vol. 2, Book 13, Hadith 1277

Regarding not supplicating for evil or the severing of blood ties - 

Jabir (radhi Allahu ahnu) narrated that the Messenger of Allah (sallallahu alayhi wasallam) said: “There is none who utters a supplication, except that Allah gives him what he asked, or prevents evil from him that is equal to it – as long as he does not supplicate for something evil, or the cutting of ties of the womb.” Classed as Sahih from Jami` at-Tirmidhi Vol. 6, Book 45, Hadith 3381

Regarding petitioning Allah by virtue of your own good deeds - 
It was narrated by 'Umar (radhi Allahu anhu) that; "I was a young unmarried man during the lifetime of the Prophet (sallallahu alayhi wasallam). I used to sleep in the mosque. Anyone who had a dream, would narrate it to the Prophet (sallallahu alayhi wasallam). I said, "O Allah! If there is any good for me with You, then show me a dream so that Allah's Apostle may interpret it for me." So I slept and saw (in a dream) two angels came to me and took me along with them, and they met another angel who said to me, "Don't be afraid, you are a good man." They took me towards the Fire, and behold, it was built inside like a well, and therein I saw people some of whom I recognized, and then the angels took me to the right side. In the morning, I mentioned that dream to Hafsa. Hafsa told me that she had mentioned it to the Prophet (sallallahu alayhi wasallam) and he said, "'Abdullah is a righteous man if he only prays more at night." (Az-Zuhri said, "After that, 'Abdullah used to pray more at night.")
Sahih al Bukhari Volume 9, Book 87, Number 156

Supplications & Treatment with Ruqyah from the Qur'an and the Sunnah - Dr Sa'eed bin Ali Wahf Al Qahtani. Darussalam 2009 Edition.