Using npm
Latest version while still respecting the semver in your package.json: npm update <package-name>.
So, if your package.json says "react": "^15.0.0" and you run npm update react your package.json will now say "react": "^15.6.2" (the currently latest version of react 15).
But since you want to go from react 15 to react 16, that won’t do.
Latest version regardless of your semver: npm install --save react@latest.
If you want a specific version, you run npm install --save react@<version> e.g. npm install --save react@16.0.0.
https://docs.npmjs.com/cli/install
Using yarn
Latest version while still respecting the semver in your package.json: yarn upgrade react.
Latest version regardless of your semver: yarn upgrade react@latest.
https://yarnpkg.com/lang/en/docs/cli/upgrade/