If you know well in advance the dates you'll be in Zion, here is all the info about the overly complicated lottery system for Angels Landing permits: https://www.nps.gov/zion/planyourvisit/angels-landing-hiking-permits.htm
But the park also keeps on hand a number of next-day tickets. If you want to hike Angels Landing tomorrow, enter the lottery today before 3pm Utah time at https://www.recreation.gov/permits/4675324.
FYI You can hike the West Rim Trail to Scout's Lookout without a permit, but if you want to continue up the famous chains section to the summit of Angel's Landing then you need a permit.