watchOS 5.3.2 can be downloaded through the dedicated Apple Watch app on the iPhone by going to General -> Software Update. To install the new software, the Apple Watch needs to have over 50 percent battery, it needs to be placed on a charger, and it needs to be in range of the iPhone.
According to Apple's release notes, the watchOS 5.3.2 update provides important security updates and is recommended for all users. A security support document says the watchOS 5.3.2 update includes a fix for a vulnerability that could allow a remote hacker to cause unexpected application termination or arbitrary code execution.
Apple has not released watchOS 6 for the Series 1 and Series 2 Apple Watch models, despite the fact that these devices will be getting the update.
Series 1 and Series 2 Apple Watch owners may need to wait for the release of watchOS 6.1 to get watchOS 6, as that beta is available for older devices as well as newer ones.
While waiting for the watchOS 6 update, Series 1 and Series 2 Apple Watch owners should download the software released today.