John Chrysostom

Zacharias glorifies the Lord, because He hath made us to serve Him with full confidence, not in the flesh as Judah did with the blood of victims, but in the spirit with good works. And this is what he means by in holiness and righteousness. For holiness is, a proper observance of our duty towards God, righteousness of our duty towards man; as, for example, when a man devoutly performs the Divine commands, and lives honourably among his fellow men. …

Catena Aurea by Aquinas ↗

Theophylact of Ohrid

For what purpose did He deliver us? Was it not so that we might live in pleasures? No, but so that we might serve Him, and not for a day or two, but every day, and serve not merely with bodily worship and service, but "in holiness and righteousness." Holiness is righteousness in relation to God, and righteousness is justice in relation to people. …

Commentary on Luke ↗

Bede

In holiness and righteousness before Him, all our days. Clearly and briefly, he designates how the Lord is to be served; namely, in holiness and righteousness, and before Him, and all our days. For he who either departs from His service before death, or pollutes the sincerity of his faith with any uncleanness or injustice, or strives to appear holy only before men and not also before the Lord; and just, is not yet perfectly liberated from the hand of spiritual enemies to serve the Lord, but, following the example of the ancient …

On the Gospel of Luke ↗

Bonaventure

And because not every manner of serving is acceptable to God, therefore he adds the manner, namely holily with respect to oneself, justly with respect to one's neighbor, simply with respect to God, perseveringly with respect to all the aforesaid. …

Commentary on Luke, Chapter 1 ↗