Is it a lack of faith if one goes to doctors?

Just the opposite! G-d wants us to bring down His miraculous healing power through these physical channels which He creates for His spiritual blessings to be manifested in the world.
As we know, G-d sends the angel of healing, Raphael (meaning “G-d has healed”) to accompany a reliable doctor. The doctor as a Divine mission to provide a healing remedy to the patient. Raphael was one of the three angels who visited Abraham after his circumcision, with the mission to bring to him the healing from G-d. He is also an angel of mercy, so from there he was sent on the mission to save Lot and his family from G-d’s destruction of the metropolis of Sodom. (See Rashi’s explanation of the verse Genesis 18:2).

The Torah-based outlook on healers and healing

Some of the greatest sages in Jewish history, for example Maimonides (Rambam), were also expert doctors. They dedicated a great amount of their time and energy to healing both Jewish and Gentile patients. The famous “Oath of Maimonides” for medical doctors reads as follows (quoted from Wikipedia):

“The Eternal Providence has appointed me to watch over the life and health of Thy creatures. May the love for my art actuate me at all time; may neither avarice nor miserliness, nor thirst for glory or for a great reputation engage my mind; for the enemies of truth and philanthropy could easily deceive me and make me forgetful of my lofty aim of doing good to Thy children.

May I never see in the patient anything but a fellow creature in pain.

Grant me the strength, time and opportunity always to correct what I have acquired, always to extend its domain; for knowledge is immense and the spirit of man can extend indefinitely to enrich itself daily with new requirements. Today he can discover his errors of yesterday and tomorrow he can obtain a new light on what he thinks himself sure of today.

Oh, G-d, Thou has appointed me to watch over the life and death of Thy creatures; here am I ready for my vocation and now I turn unto my calling.”

Take your medicine with a prayer

There are some short prayers that are traditional to say before and after taking medication. From the Mishneh Brurah, this prayer should be recited BEFORE taking medicine or when one undergoes a medical procedure:

“May it be Your will L-rd G-d, that this use of medicine bring me good health, for You heal graciously.” [Code of Jewish Law, Orakh Hayim 230:4]

AFTER taking medicine or after a medical procedure, it is traditional to say:

“Blessed is the Healer of the sick.”