Synchronizing manually on iOS is easier than ever. While in any type of list, simply drag the screen downwards and you'll see a spinning loading wheel. After that, you should see a little message at the top of the screen to tell you that your list is synchronized! Note that even after you have manually synchronized, it can still take a few minutes for the data to appear in your account on a different device (or on the web). In addition, this type of sync works for your whole account - not just for the list from which you are working. While the app also synchronizes on its own in response to a number of user actions, some users (particularly those shopping from a shared list while others shop simultaneously) like to manually sync as a way to speed up the process.
Here's a GIF to show you what the spinning wheel looks like: