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 Importance of Visiting the Sick.

The Prophet (sallallahu alayhi wasallam) went to visit a sick bedouin. Whenever the Prophet went to a patient, he used to say to him, "Don't worry, if Allah will, it will be expiation (for your sins):"... 

Reported in Riyad as-Salihin in The Book of Visiting the Sick, the Hadith being from Sahih Al Bukhari Book 7, Hadith 907.

In this benevolent prophetic saying, there is an expression of sympathy, a most noble show of compassion, as well as a most favourable type of treatment. Such attitude on the part of the visitor will relieve the patient’s anxieties, refresh his spirit and animate his heart. Subsequently, it will help change his natural disposition and aid his nature to overcome the effects, and sometimes lessen, or even alleviate his symptoms and defeat the cause. This is in effect, the purpose behind calling a doctor.

Visiting the sick is one of the 7 things enjoined upon those who believe.

It was narrated by Al-Bara' bin 'Azib (radhi Allahu anhu) that the Prophet (sallallahu alayhi wasallam) ordered us to do seven (things) and forbade us from seven. He ordered us to visit the patients, to follow the funeral procession, to reply to the sneezer (i.e., say to him, 'Yarhamuka-l-lah (May Allah bestow His Mercy upon you), if he says 'Al-hamdulillah' (Praise be to Allah), to help others to fulfill their oaths, to help the oppressed, to greet (whomever one should meet), and to accept the invitation (to a wedding banquet). He forbade us to wear golden rings, to use silver utensils, to use Mayathir (cushions of silk stuffed with cotton and placed under the rider on the saddle), the Qasiyya (linen clothes containing silk brought from an Egyptian town), the Istibraq (thick silk) and the Dibaj (another kind of silk).

Sahih Al Bukhari Volume 7, Book 62, Number 104

The beloved Holy Prophet (sallallahu alayhi wasallam) showed great love and compassion when he visited sick persons, particularly his companions (radhi Allahu anhum). He would gently ask about their point of discomfort, their condition and present feelings. He cheered them up, placed his blessed band over their foreheads and sometimes placed it in the middle of their chests, and prayed for them. He (sallallahu alayhi wasallam) also prescribed the correct medicine for them, and in one case it is reported that the Messenger of Allah (sallallahu alayhi wasallam) would take an ablution and wash the patient’s face or body with the water of that ablution or sprinkle the water upon them. This is the utter kindness, the noblest treatment and the correct mannerism to demonstrate with those that are sick.

Sympathic-therapy sometimes does wonders for curing an illness or at least, lessening its effects. Often, this kind of support helps the human spirit, builds up a patient’s will and empowers the psycho-physiological process in his mind to defeat the disease.

Visiting the sick offers four kinds of benefits;

• Mental benefits for the patient that helps reconstruct their state of mind and natural bodily healing processes and livens their spirits.
• Benefits for the visitor in the form of blessings and forgiveness from Allah Almighty and the reward for following the Sunnah of Rasool'Allah (sallallahu alayhi wasallam).
• Conciliatory benefits for the nursing family and the strengthening of ties and good manners.
• Constructive benefits for the Muslim community in that local groups take care to support each other.