Magento 2 – Adding a jQuery Library

Likely you are having a conflict with prototype.js. You can do as Marius pointed out and use prototype which is already used throughout Magento, or if you'd like to use jquery, you can use a noconflict. You could also just replace you '$' with 'jQuery' to avoid conflict.

<script type="text/javascript">
    jQuery("#Category1").click(function(){
        jQuery(".moreCategory1").toggle();
    });
</script>

You have to set jQuery dependency for your custom script. You have occurance of problem because of dependency issue.

1. For Module/Extension Level

You can set at module/extension level dependency like below,

app/code/Packagename/Modulename/view/frontend/requirejs-config.js

var config = {
    paths: {            
            'slider': "Packagename_Modulename/js/slider"
        },   
    shim: {
        'slider': {
            deps: ['jquery']
        }
    }
};

Add slider.js inside path, app/code/Packagename/Modulename/view/frontend/web/js/slider.js

You can use in any template file like this,

<script>
    require(["jquery","slider"],function($)(){
        $(document).ready(function() {
            $(element).slider({
                   //option
            });
        });
    });
</script>

2. Theme Level

If you want to apply require js changes at theme level, Go to theme module,

app/design/frontend/{Packagename}/{themename}/Packagename_Modulename/requirejs-config.js

Set your theme required js file in requirejs-config.js

var config = {
    paths: {            
            'slider': "Packagename_Modulename/js/slider",
            'customjsfile': "Packagename_Modulename/js/customjsfile",
        },   
    shim: {
        'slider': {
            deps: ['jquery']
        },
        'customjsfile':{
            deps: ['jquery']
        }
    }
};

Add your js file inside web folder of theme,

app/design/frontend/{Packagename}/{themename}/Packagename_Modulename/web/js/slider.js
app/design/frontend/{Packagename}/{themename}/Packagename_Modulename/web/js/customjsfile.js

In your template file,

<script>
    require(["jquery","slider","customjsfile"],function($)(){
        $(document).ready(function() {
            $(element).slider({
                   //option
            });
        });
    });
</script>

For your case,

Keep below config in requirejs-config.js file,

var config = {
    paths: {            
            'footerContact': 'Module_Contact/js/contact'
        },   
    shim: {
        'footerContact': {
            deps: ['jquery']
        }
    }
};

Inside phtml file,

<script>
  require(["jquery","footerContact"],function($)(){
    jquery('.element').on('click', function(event) {
        console.log('logging');
    });
  });
</script>