Once you've clicked the Notes app icon and you have a new notes file open, locate and tap the "+" icon that can be found on the bottom middle of your screen. If your keyboard is on, it will appear in a bar right above the keyboard.
Note: If you don't see it, go to Settings->iCloud->Notes and turn that on. Then, once you go back to Notes, the "+" should appear.
Image Source: POPSUGAR / Brittany Natale