Based on the American Meteor Society, the Draconids will peak Tuesday night into Wednesday morning, and the South Taurid meteor shower will peak Wednesday night into Thursday morning.
The Draconids shower will peak within the early evening to nightfall, which means you can catch this one easily before midnight.
There is a full moon on October 13, and in the days up to then, the moon could obscure some of the light from the meteor showers.
Debris from the comet 21P/Giacobini-Zinner is liable for the Draconid meteor shower, which will get its title because the meteors look like coming from the direction of the constellation Draco the Dragon.
Based on Earth Sky, which monitors meteor showers:
Draco the Dragon is now expelling meteors also called shooting stars. That is one shower that’s best to observe in early evening and nightfall, but not after midnight. No matter where on Earth, watch as near to nightfall as possible. The shower is active between October 6 and 10. One of the best evening to watch is likely October 8; attempt the evenings of October 7 and 9 additionally. This shower favors the Northern Hemisphere, however, Southern Hemisphere observers may catch some Draconids, too. Unfortunately, the large bright evening moon will hinder this year’s Draconid shower. It’ll likely drown all, however, the brightest meteors in its glare.
The South Taurid meteor shower will last by November, averaging about five meteors per hour. The full moon will also affect seeing this shower; however, a new moon later in the month should provide more views.
Debris from Comet 2P/Encke generates The South Taurid meteor shower.