Los Angeles proper has never had its own mission (200+ years ago, the San Gabriel and San Fernando missions were about a day's journey from the pueblo, which almost makes modern LA traffic seem less atrocious). Most people don't know that.
In many cities, the oldest building is likely to be a house of worship. There is some debate over what LA's oldest building is, but La Placita, or the Old Plaza Church, is certainly one of the oldest.
Before the church's dedication in December 1822, the pueblo's residents had to rely on visiting priests from the San Gabriel and San Fernando missions to have their spiritual needs met (non-Catholics began to arrive in the 1840s).
La Placita's first resident priest, believe it or not, was a Frenchman. Jean-Augustin Alexis Bachelot was born and educated in France, becoming a priest in 1820 at the age of 24. In 1827, he led the first Catholic mission to Hawaii (besides Catholicism, Bachelot introduced bougainvillea* and mesquite plants to the islands). Bachelot and his fellow priests were well-received by locals (the fact that Bachelot learned the Hawaiian language well enough to translate a prayer book and write a Hawaiian-language catechism probably helped), but they faced persecution by the Protestant regent, Queen Ka'ahumanu, who deported them from Hawaii in December 1831.
Bachelot and fellow priest Patrick Short landed near San Pedro in January 1832, traveling to Mission San Gabriel. Not only did Bachelot become La Placita's first resident priest, he served as the mission's assistant minister, temporarily led the mission when its head priest was reassigned in 1834 (turning down the substantial salary he was offered), and taught in one of LA's first schools during a teacher shortage. Besides French and Hawaiian, Bachelot spoke Spanish well (the Autry Museum of the American West has, in its collection, a photostat of a letter Fr. Bachelot wrote in 1836 - in Spanish). He was, by all accounts, well-liked by Angelenos.
Bachelot ministered in Los Angeles until 1837, when he had the opportunity to return to Hawaii. Sadly, things did not go well in Hawaii (see above), his health suffered, and he passed away later that year while at sea. Bachelot was buried off the coast of Pohnpei, Micronesia. Because of the way their priests had been treated in Hawaii, the French government intervened, and King Kamehameha III finally granted religious freedom to Catholics in Hawaii.
In Los Angeles, Bachelot was succeeded by another French priest - Reverend Anaclet Lestrade. Like Bachelot before him, he doubled as a teacher - in 1852, he taught twenty students due to lack of a proper school system (public, private, or parochial - LAUSD didn't exist until 1853). Lestrade is credited with helping to establish the first boys' boarding school in Southern California. For a time, he also held claim to the Rancho Rosa Castilla in El Sereno.
As for the building itself...the original church was destroyed due to severe flooding in 1859-1860 (the LA River burst its banks a few times - which is quite a thing to contemplate if, like me, you have only ever seen it as a tiny trickle in a vast concrete ditch). La Placita was rebuilt under the leadership of LA's popular French-Canadian mayor, Damien Marchesseault (more on him later...stock up on tissues, it's a sad story).
Henri Penelon, LA's first commercial artist and photographer, painted a mural of the Madonna and Child with two angels over the door in 1861, assisted by a new arrival, 21-year-old Bernard Etcheverry (both hailed from France, and don't worry, you'll read more about them later). Sadly, the mural - probably the first outdoor mural in Los Angeles - is long gone. In yet another example of Los Angeles erasing its own history, the mural was plastered over in 1950 (there is now a mosaic on that part of the facade, installed in 1981). The church's marble tablets bore Penelon's lettering at least as late as 1932, and the Stations of the Cross are consistent with his other work.
Although a bit beyond the topic, but worth noting, are La Placita's bells. They were cast by George Holbrook, apprentice to Paul Revere - whose father was a French Huguenot.
Let's not forget the surrounding neighborhood. In Fr. Bachelot's day, French transplant Pierre Domegue and his wife (a Chumash woman named Maria Dolores Chihuya) baked French-style sourdough bread in a low adobe that stood next to the church's courtyard. Domegue also partnered with another French baker, Andre Mano, in a bakery just around the corner (Angelenos weren't afraid of carbs yet).
La Placita is still an active parish church. Do take the time to visit, but please be respectful of those who are there for spiritual reasons.
*Edited to add (7/1/17): A friend who reads this blog lived on Kauai for 25 years. She told me that Fr. Bachelot brought bougainvillea plants to Hawaii because they're thorny, and the Catholic Church was trying to get Native Hawaiians to wear shoes (and, for that matter, other Western garments). She describes Bachelot as "a bad, bad man" and tells me he's widely disliked in Hawaii. In the interest of presenting history in a fair and truthful manner (Native Hawaiians' stories matter too), I felt I should add the dark side of Bachelot's story. (And honestly, it sounds exactly like something the Church would have done back then.)