How can I find ancient cores in Breath of the Wild?
According to IGN's Wiki, they can be found on the following:
- Guardian Stalkers
- Guardian Skywatchers
- Guardian Turrets
- Sentries
- Guardian Scout III
- Guardian Scout IV
They cannot be found at Decayed Guardians. There are also Shrine chests that contain them:
- Treasure Chest in Akh Va'quot Shrine in the Tabantha Region
- Treasure Chest in Daag Chokah Shrine in the Woodland Region
- Treasure Chest in Dah Kaso Shrine in the Central Hyrule Region
- Treasure Chest in Dako Tah Shrine in the Gerudo Wasteland Region
- Treasure Chest in Hawa Koth Shrine in the Gerudo Wasteland Region
- Treasure Chest in Ishto Soh Shrine in the Lake Hylia Region
- Treasure Chest in Kaya Wan Shrine in the Lanayru Region
- Treasure Chest in Ketoh Wawai Shrine in the Woodland Region
- Treasure Chest in Maka Rah Shrine in the Hebra Region
- Treasure Chest in Rin Oyaa Shrine in the Hebra Region
- Treasure Chest in Shai Utoh Shrine in the Faron Region
- Treasure Chest in Tu Ka'loh Shrine in the Akkala Region
The other two answers here are good if you're sticking within the game, but if you're willing to shell out for an Amiibo, the guardian Amiibo drops random guardian parts, as confirmed by this source. Several other sources from a quick googling confirm this (I can't personally confirm it, they were all sold out when I was playing BotW) (I can personally confirm it now that I own the amiibo). I can't find a source saying what percentage of the random drops are an ancient core, but you can reuse the Ammibos each day, so if you're planning on spending several days farming ancient parts, this will help quite a bit.
As explained on this page, ancient cores only drop from the walking or Flying Guardians, the Sentry Guardians (in Hyrule Castle), and the Scout Guardians in the Major Test of Strength shrines.
I have found that the Scout Guardians in the shrines have the highest drop rate. I would estimate about a 30-50% drop rate, based on my own experience. Since these Guardian types respawn after every Blood Moon, this makes farming ancient cores much less aggravating. Sentry Guardians also drop cores with a decent drop rate, but they are much more dangerous - I would recommend getting some Ancient Arrows and a Lynel bow to defeat them.