Continuity features need your devices signed in to the same Apple Account with Wi-Fi and Bluetooth turned on, and close to each other. Most of the settings are in System Settings > General > AirDrop & Handoff.
1Handoff
Turn on Allow Handoff between this Mac and your iCloud devices in AirDrop & Handoff (and Handoff on your iPhone or iPad). Start an email, web page or document on one device, and an icon appears in the Dock of the other. Click it to carry on where you left off.
2AirDrop
Open AirDrop from the Finder sidebar, or use Share > AirDrop in any app. Golden Gate makes AirDrop transfers faster. Set who can send you files in AirDrop & Handoff or Control Centre: Contacts Only is the sensible default; Everyone for 10 Minutes is there for one-off transfers. Received files go to Downloads.
3Universal Clipboard
With Handoff on, copy on one device and paste on another within a short time. It works for text, images and files. No setup beyond Handoff is required.
4Sidecar
Use an iPad as a second display: open System Settings > Displays, click + (or the Add display menu), and pick your iPad. In Golden Gate, Sidecar has much more complete touch support, including tapping controls, one-finger scrolling, system gestures and Markup, so you can use the iPad as a touchscreen for your Mac. An Apple Pencil also works.
5iPhone Mirroring
Open the iPhone Mirroring app on the Mac to see and control your iPhone, with its notifications appearing on the Mac. In Golden Gate the window can be resized to different sizes and aspect ratios, which is useful on large, high-resolution displays (up to 5K; check Apple Support for exact requirements). Your iPhone stays locked while you use it.
iPhone Mirroring isn't available in the European Union. It needs a compatible iPhone on a current iOS version, with the iPhone nearby and locked.