What is thought to be the first anime was a 3 second clip that is 50 frames long, called Katsudo Shashin. In it, a boy writes 活動写真, which means "moving pictures" on a board. Its creator and date of creation are unknown, but it is thought to be from circa 1907.
There is an anime short from June 1917 named なまくら刀 (An Obtuse Sword) that is 2 minutes long and a 1918 film adaptation of Urashima Taro, or 浦島太郎 that are thought to be the first films. The Kyoto International Manga Museum cites Imokawa Mukuzo Genkanban no Maki from April 1917 as the first Japanese produced animation, though the Wikipedia article implies that there were previous works.
The first color anime feature film, which is sometimes considered to be the first anime by modern standards, is Hakujaden, which was created in 1958.
おとぎマンガカレンダー, or Otogi Manga Calendar, was the first anime series to be produced and the first to be televised. It ran from 1961-1964. Astro Boy first aired in 1963, making it one of the older anime series, but not the oldest. However, Astro Boy was the first 30 minute anime series as well as the first anime series that aired weekly.
Manga is a little bit more complicated because comics existed and it's hard to draw the line between where something stops being a comic and begins being a manga. Sazae-san was a comic strip published from 1946-1974 that has the longest ever anime series based off of it. Astro Boy is more generally referred to as a manga, and it began in 1951. There doesn't seem to be one specific manga cited as the first manga, though it seems to often be consider to have begun after WWII, as explained in this.
Edit: To add to this, Momotaro: Umi no Shinpei (桃太郎 海の神兵) was the first feature-length anime, which means that, at 74 minutes, it passed the 40 minute mark to be considered feature-length.
Edit 2: Some new information has been added based on information in the Kyoto International Manga Museum