Modems and Routers
Let's start at the beginning: there a wire (cable or DSL or fiber or what have you) coming from the street to your house, and that wire connects to a modem. That modem can have a router built-in, or you might have a separate router, for instance an AirPort base station. Routers are used to allow multiple devices to operate on your internet connection, and most include Wi-Fi for wireless access and internal networking. From the router, internet access is “routed” to devices, making sure all the web pages you ask for on your iMac and the Apple TV streaming Netflix video each get the right data.
What? I cleaned it up so it looked nice for you.
Meet your equipment
What kind of cable modem do you have? What model is your smart garage door opener? When things are humming along it's easy to not care about that information but when it comes apart, knowing what model you have is important.
At my house we have a shared Dropbox folder that has a list of what all the internet capable devices are, with the make and model, and their manuals. (With Dropbox, at least if nothing else is working it can be pulled up on an iPhone by whoever needs it.)
A lot of product manuals can be found online, so if you aren't sure what the “normal” indicators are or need information about your hardware, it's a good idea to grab that manual now while things are working. It's a lot nicer to have a digital copy anyway since you can easily search PDFs for an exact phrase.
This is about half of the folder. I download the manuals for ALL THE THINGS.
Other (Working) Services
In 2015, most people are getting their internet from the same provider as their television service, phone service, or both. So if your internet-connected devices aren't working, start at the street and work your way into the house. If you don't have internet, see if you have television or phone service. They aren't always related, but if none of the services are functional, it's better to find out before doing a bunch of troubleshooting that turns out to be pointless.
If you still have a working phone, try calling your service provider. In a lot of cases, I have called and received an automated message that something was down in my neighborhood, which is frustrating but not as bad as spending half an hour working on the problem with your devices only to find out there's nothing you can do.
First/Second Rules
If it seems like there are no external forces thwarting you, it's time to start troubleshooting. As I have mentioned before, the first and second rules of tech support are “is it plugged in?” and “is it turned on?” You might be surprised how often these two questions cut right to the heart of tech problems. There's a reason these are the top two rules.
A good way to double check this is unplug any relevant cables entirely and plug them back in, just to be sure. If there are any sort of indicators (like status lights for power and such), make sure they're responding properly.
This happens a lot more often than you'd think.
Identifying the Failure
Even if your internet is down, any items that don't “phone home” (require access to a remote server) should still work fine and be controllable while you're on the same Wi-Fi network. Example: I can still control my Hue lights while I'm home because we're talking over the same network. If it's just the internet that's down, things in your house can still talk to each other, just not anything else. But if there's a problem at the modem or out to the street and your provider, I wouldn't be able to talk to my Hue lights away from home.
If nothing in your house seems to be controllable, or your iPhone remote app can't get to the Apple TV anymore, it might be your router that is the problem. Try unplugging the power to the router for a moment and plugging it back in. Don't leave it unplugged too long—with some brands, special configurations can be lost.
If you only have an issue on one device, then it might be the device that's suspect. Have some idea how to “reboot” that device, whether it's a Hue lights, or WeMo units, or door locks or anything else. Sometimes it's as easy as unplugging and plugging in the power, but in some cases that isn't the best way to restart a device, so make sure you know (or have access to instructions on) how to do this for your gear.
Even with no internet, I can still run all my Hue lights.
Next steps
Once you find the problem, how to solve it ranges from looking at the troubleshooting in the back of the manual (which you totally already have thanks to downloading it) to discovering that your router has run off and joined the choir invisible. I don't have room to list all the available solutions here, I don't even have room to list all the things that have worked for me! But hopefully with the list above you will be able to at least have a clear idea what the problem is so you can get it taken care of with a minimum of hassle.