Here are a few suggestions. If you've done everything I've mentioned here you should be decently secure.
AndroidOpen settings, then open security and privacy. On my phone I see "Looks good. No security issues". If you don't see this, then look areas that don't have green check marks and do whatever the phone wants you to do to make them green.

You also want to make sure you have a password on your phone. Don't use a 4 digit pin or a short pattern. If you are going to use a pattern lock then make sure it's longer than the default.
Depending on the model of android, the phone might have full disk encryption already enabled(mine does). If it doesn't you should see an option that says something like "full disk encryption" or "file encryption" or something with encryption. If you can find this option and it's off, turn it on. If your disk isn't encrypted, then someone can plug your phone into a computer and then browse the files on your phone as if it were a USB harddrive.
AppleThe same password suggestions from above apply here.
I don't use apple devices, but your phone should be quite secure out of the box as long as you use a screen password. The main way someone is going to get past your screen password is going to be by watching you type it in, which is why you want to use something longer than a 4 digit pin.
Suggestions for your browser:Find the option in your browser to enable HTTPS everywhere. This will ensure that when a website has the option to encrypt your web traffic it will.