Some people think their iPhone apps are only "organized" when they're in a certain order from left to right, or top to bottom. I disagree. The best way to arrange your apps is to put the ones you use the most within easy reach of your fingers. 
As for how to deal with the rest of your apps, I have a few strategies you can use to arrange them in a way that makes sense while also keeping your screens neat. When relevant, there's a brief how-to next to the tip. These tips also work for the iPad, though finger-reach is usually a more pressing issue on smaller devices.
1. Put Favorite Apps in the Dock
The app dock is the bottom row of apps on your iPhone. When you swipe between screens of apps on your phone, the dock stays put. This is where you should have icons for the apps you use the most. The dock holds up to four apps or folders. It cannot hold any more unless you jailbreak your phone, which I don't recommend for most people.
How to Move iPhone Apps: For true iOS beginners, here's how to move app icons. There are two methods. First, press and hold any app until a selection box appears. Quickly lift your finger and choose Edit Home Screen. (It says this for all the following icon pages, not just the first.) The second way goes like this: Press and hold any app, and when that selection box appears, ignore it. Keep holding. All your apps will soon jiggle, meaning they're unlocked from their places and you can move them.
To move an app, press your finger on it and slide it around the screen. When the app is where you want it, lift your finger. The app locks into place. To lock them all down, tap Done or the home button, depending on which phone you have.
How to Put Apps Into the Dock: To put an app icon in the dock, there must be a free space for it. If there is, follow the previous instructions and slide it into place. That's it.
To make space in your dock, simply move an app out of the dock. Or delete an app; when the apps are jiggling, tap the X that appears on it. Apps without an X cannot be deleted. When finished, tap Done or the home button.
How to Put a Folder Into the App Dock: First, make a folder of apps (see instructions below). Then, free up a space in the dock. Finally, drag the folder into the dock. Release it and lock it into place by tapping Done or the home button.
2. Put Frequently Used Apps Along the Edges
For most people, the bottom of the phone is within easy reach of your fingers. That's why the dock is there and why other quick-action buttons come from swiping up from the bottom. The second-most-used areas are corners and edges. Use these zones to keep the apps you use often but which didn't make it into the dock.
If you're right-handed and use your thumbs to navigate, you want to utilize the lower-right corner and right edge. Lefties have it reversed. I hold my phone in my left hand and use the middle finger of my right hand to tap and swipe, making my high-use area the bottom two rows of apps.
3. Cluster Similar Apps in the Middle
The third tip for organizing your iPhone apps is to cluster together certain kinds of apps in the middle of the screen. It's an ideal place for apps that you use often, but that you don't need to reach as easily. It's also handy for apps that you use in tandem or together in a workflow.
For example, if you tend to browse images in your Photos app, then edit them in a photo-editing app, and finally post them to Instagram, it makes sense to place those three apps together. Another example is ride-sharing apps: I have a small cluster of them on my phone because if I have a hard time getting a car in one app, I want to be able to see my other options quickly.
By now, you're probably thinking about which apps to keep on the first screen of your iPhone and which ones to stick on screens farther back. For anyone unfamiliar with how to make new screens, here's a quick tutorial.
How to Make New App Screens: Once you fill up your first screen with apps, the iPhone automatically puts any more apps you download onto subsequent screens. You can have up to 15 screens (or 11 with older versions of iOS). You can make new screens, however, before the current one fills up by pressing and holding any app and dragging it to the far right until a new screen appears. Lift your finger to drop the app, and tap Done or the home button to lock everything into place.
4. Use Folders for App Groups
Folders are handy for grouping apps by theme or usage. Apps that are in folders are among the least easily accessible, making them a great place to store apps that you need but don't use often. You can have up to 135 apps in a folder (which, frankly, seems excessive).
One way to organize apps into folders is by theme. Some of my themed folders are food delivery, personal finance, and Apple apps.
Another way to group apps into folders is by how or when you use them. Let's say you travel for business. You could have a folder containing your preferred airline's app, some for your favorite hotels, and an app for scanning receipts and creating expense reports.
How to Make a Folder in iOS: To create a folder, press and hold any app until they all jiggle. Hold and slide any app on top of any other app until the iPhone automatically creates a folder. Release your finger. You'll see that iOS suggests a name for the folder, which you can change by tapping the text field at the top and typing a new one. Press anywhere outside the folder to return to the previous screen and to continue sliding more apps into the folder. When you're done, tap Done or the home button.
5. Leave Blanks and Push Some Apps to the Back
I mentioned earlier that you can create a new screen before the current one is full. Use this to your advantage! You don't have to fill every available slot on every screen. Drag apps to the right to create a new screen, and leave them there if you don't want to look at them for a while.
It's easy to forget about those apps you send to far-off screens, and there's a setting in iOS that will actually make those apps disappear if you don't use them often enough and your phone gets tight on space. It's called Offload Unused Apps. I recommend disabling it, in part because I have yet to see any clear information regarding how infrequently the app must be used and how tight on space your device must be before these apps go away. Here's how to disable it.
How to Disable "Offload Unused Apps" in iOS: Go to Settings > tap your name at the top (it also says Apple ID, iCloud, iTunes & App Store) > iTunes & App Store > and toggle the button next to Offload Unused Apps.
Confusingly, there's a second place where you can enable it, but not disable it. That one's at Settings > General > iPhone Storage > Offload Unused Apps.
Make It Yours
When organizing your iPhone app icons, do what works for you. Put the apps you use the most in spots that are easy to reach. Organize other apps according to how often you use them, how easy or hard it is to reach them, and whether you use those apps alone or with others.
For more iOS tips, see How to Free Up Space on Your iPhone or iPad and How to Save Battery Life on Your iPhone.