How can I save some user data locally on my Xamarin Forms app?

You have a couple options.

1. SQLite. This option is cross-platform and works well if you have a lot of data. You get the added bonus of transaction support and async support as well. EDIT: In the past I suggested using SQLite.Net-PCL. Due to issues involving Android 7.0 support (and an apparent sunsetting of support) I now recommend making use of the project that was originally forked from: sqlite-net
2. Local storage. There's a great nuget that supports cross-platform storage. For more information see PCLStorage
3. There's also Application.Current.Properties implemented in Xamarin.Forms that allow simple Key-Value pairs of data.

I think you'll have to investigate and find out which route serves your needs best.

As far as security, that depends on where you put your data on each device. Android stores app data in a secure app folder by default (not all that secure if you're rooted). iOS has several different folders for data storage based on different needs. Read more here: iOS Data Storage

Please use Xamarin.Essentials

The Preferences class helps to store application preferences in a key/value store.

To save a value:

Preferences.Set("my_key", "my_value");

To get a value:

var myValue = Preferences.Get("my_key", "default_value");

Another option is the Xamarin Forms settings plugin.

E.g. If you need to store a user instance, just serialize it to json when storing and deserialize it when reading.

Uses the native settings management

• Android: SharedPreferences
• iOS: NSUserDefaults
• Windows Phone: IsolatedStorageSettings

• Windows RT / UWP: ApplicationDataContainer

    public User CurrentUser
    {
       get
       {
          User user = null;
          var serializedUser = CrossSettings.Current.GetValueOrDefault<string>(UserKey);
          if (serializedUser != null)
          {
             user = JsonConvert.DeserializeObject<User>(serializedUser);
          }
  
          return user;
       }
       set
       {
          CrossSettings.Current.AddOrUpdateValue(UserKey, JsonConvert.SerializeObject(value));
       }
    }
  

EDIT: There is a new solution for this. Just use Xamarin.Essentials.

Preferences.Set(UserKey, JsonConvert.SerializeObject(value));
var user= JsonConvert.DeserializeObject<User>(Preferences.Get(UserKey, "default_value");