Handoff and Continuity are the biggest gee-whiz features in OS X Yosemite and iOS 8. The ability to start composing an email, view a web page, or work on a document on one device, then switch to another, could actually merit the use of the word "magical," which Apple has applied to a number of its new products.
Except when it doesn't work.
In the month Yosemite has been out, I've gone through an arduous process to get Handoff, Continuity, and even AirDrop to work. While the features work most of the time, they don't always work, and still not on all of my devices. The exceptions so far have been phone calls and SMS forwarding, which worked from the beginning--when my iPhone rings, my office resounds in a symphony of asynchronous ringing from my various Apple devices.
Are you compatible?
The first thing to do when troubleshooting problems with new features like these is check whether your devices are compatible. In this case, all the supported devices have Bluetooth LE, which, as Chuck La Tournous recently explained, "will only connect devices that use the same Apple ID, signed into iCloud. Only then will the Handoff features be enabled." In addition, Bluetooth and Wi-Fi must be on, on all your devices, and they must be connected to the same Wi-Fi network.
As for me, I've got five compatible devices: My Retina 5K iMac, 2013 Retina MacBook Pro, iPhone 5s, iPad Air 2, and fifth-gen iPod touch.
The logout dance
At first, when I was having problems getting Handoff and Continuity to work, I tried what seems to be the standard troubleshooting tip: logging each of my devices out of iCloud, then logging in again.
To do this on a Mac, go to System Preferences > iCloud, and click Sign Out. This is a complex process, with your Mac asking if you want to keep your contacts, calendars, and so on. You should also make copies of any files you have in iCloud Drive, if you're using that feature.
On an iOS device, you can do the same thing from Settings > iCloud; scroll down and tap Sign Out at the bottom of the window, and follow the instructions.
When I did this shortly after iOS 8.1 was released, nothing changed. I spent a long time on the phone with a senior AppleCare technician, who felt that the only possibility was that my router was blocking some sort of connection. For these features to work, your devices have to communicate with iCloud servers to be able to share the data about files, web pages, and so on. Strangely, this seems to be independent from iCloud Tabs in Safari, which has worked reliably for me.
I examined my router's settings, turned off its firewall, and disabled a few other features, but no dice; I was still handoffless and discontinuous.
A few days later, I tried the iCloud login dance again. I logged every one of my devices out of iCloud, then logged them in again. Suddenly, Handoff was starting to work, but not every time. Safari web pages were showing up on other devices, though not on all of them. Emails were being handed off, but, again, not on all my devices.
The iCloud Drive catch
There's an extra gotcha when using iWork apps. Handoff only works if the files you're working on are saved to iCloud Drive. So if you're not getting those apps to work, choose File > Move To, and move your files to iCloud Drive.
After all this work, however, my MacBook Pro still won't work. I've signed out of iCloud and signed back in again, and it's a compatible model. At this point, I've pretty much given up; these features seem not to be worth jumping through this many hoops to get them to work. I do use them to view web pages in Safari, say, first on my iPad, then on my Mac; or to start replying to an email on my iPhone, then switching to my Mac when I realize I need to attach documents. But other than that, I feel like I've done more than I should for little return.
No good solution
Apple really needs some sort of troubleshooting tool to help users find where things are going wrong--similar to the Connection Doctor in Mail, perhaps. Apple has spent a lot of time marketing these features, so the compay owes its users better support. Apple devices used to "just work," and these features just don't.