1. First of all, make a backup
However reliable the transfer tools are, it's worth always having a recent backup of your old phone before starting the switch. On Android you can turn on the backup at Settings → Google Account → Backup, and on iPhone at Settings → [your name] → iCloud → iCloud Backup. With that backup done, any of the methods below is much safer: if something fails halfway through, you lose nothing.
2. Transferring data from one Android to another Android
If you're switching from one Android to another Android, it's the simplest transfer of all. When you turn on the new phone for the first time, the setup assistant offers to copy the data from the old one using Google's tool. You can do it with a cable connecting both phones (faster) or over wifi if you don't have a cable handy. Just bring the two phones close together, accept the transfer and wait for contacts, photos, apps, settings and saved passwords to be copied. If you'd already finished the initial setup, you can repeat the process from Settings → System → Backup and phone switching.
3. Transferring data from one iPhone to another iPhone
Between two iPhones, Apple uses the Quick Start feature. Just turn on the new iPhone near the old one: a prompt will appear on the old phone's screen asking if you want to use it to set up the new one. After scanning an animation with the camera and entering your passcode, both devices connect and you can choose to transfer the data directly over wifi or restore from your latest iCloud backup. The process includes photos, contacts, apps, settings and even the passwords saved in iCloud Keychain.
4. Moving your data from Android to iPhone
If you're switching from Android to iPhone, Apple has an app designed exactly for this: "Move to iOS", available on Google Play. Download it on your Android before starting the setup of the new iPhone, and when it asks you at the "Apps & Data" step, choose "Move Data from Android". The app generates a code you enter on your Android to link both phones over wifi, and from there contacts, message history, photos and videos, browser bookmarks and email accounts are copied. The equivalent apps for your favorites are downloaded automatically if an iPhone version exists.
5. Moving your data from iPhone to Android
The other way around, Google offers the "Switch to Android" app for iPhone, available on the App Store. When setting up the new Android, the assistant will offer to transfer data from an iPhone and guide you through connecting both phones, usually by scanning a QR code with the iPhone's camera. The tool copies contacts, photos, videos and calendar, and looks on Google Play for the Android versions of the apps you already had installed so you can download them with a single tap.
6. How to transfer WhatsApp to the new phone
WhatsApp needs its own step because chats don't travel the same way as the rest of the data. If you stay on the same operating system (Android to Android or iPhone to iPhone), it's enough to have a chat backup turned on beforehand: when you install WhatsApp on the new phone and sign in with your number, the app detects the backup and offers to restore it automatically. If you're switching systems, between Android and iPhone, WhatsApp includes its own "Transfer chats to Android/iOS" feature: you need to start the migration from the old phone following the on-screen instructions, keeping both phones close together and connected throughout the whole process until it finishes.
7. What to do with the old phone before selling, donating or recycling it
Once you confirm everything has copied correctly to the new phone, deleting the apps from the old one isn't enough: you need to properly sign out of your accounts and remove your trace. On Android, go to Settings → Accounts and remove your Google account, then run a factory reset from Settings → System → Reset options. On iPhone, first unlink the device from Settings → [your name] → Find My → Find My iPhone (or directly by doing "Erase All Content and Settings" from Settings → General → Transfer or Reset iPhone). In both cases, remove the SIM card and the memory card first if it has one. This leaves the phone clear of personal data and ready to sell, donate or take to a recycling point.