Primary and Secondary Button Classes

It looks like you’re trying to use classes from the WordPress admin styles on the front end. However, those styles are not loaded. You have two options, one of which I wouldn’t recommend.

Way #1: Do this

What I would recommend is that you simply copy and paste the buttons styles you want into your child theme or custom CSS plugin (example, one of many).

Here’s a first pass at pulling them out and cleaning them up:

.button,
.button-primary,
.button-secondary {
    display: inline-block;
    text-decoration: none;
    font-size: 13px;
    line-height: 26px;
    height: 28px;
    margin: 0;
    padding: 0 10px 1px;
    cursor: pointer;
    border-width: 1px;
    border-style: solid;
    -webkit-appearance: none;
    -webkit-border-radius: 3px;
    border-radius: 3px;
    white-space: nowrap;
    -webkit-box-sizing: border-box;
    -moz-box-sizing: border-box;
    box-sizing: border-box;
}

.button,
.button-secondary {
    color: #555;
    border-color: #cccccc;
    background: #f7f7f7;
    -webkit-box-shadow: inset 0 1px 0 #fff, 0 1px 0 rgba(0,0,0,.08);
    box-shadow: inset 0 1px 0 #fff, 0 1px 0 rgba(0,0,0,.08);
    vertical-align: top;
}

p .button {
    vertical-align: baseline;
}

.button:hover,
.button-secondary:hover,
.button:focus,
.button-secondary:focus {
    background: #fafafa;
    border-color: #999;
    color: #222;
}

.button:focus,
.button-secondary:focus {
    -webkit-box-shadow: 1px 1px 1px rgba(0,0,0,.2);
    box-shadow: 1px 1px 1px rgba(0,0,0,.2);
}

.button:active,
.button-secondary:active {
    background: #eee;
    border-color: #999;
    color: #333;
    -webkit-box-shadow: inset 0 2px 5px -3px rgba( 0, 0, 0, 0.5 );
    box-shadow: inset 0 2px 5px -3px rgba( 0, 0, 0, 0.5 );
}

.button-primary {
    background: #2ea2cc;
    border-color: #0074a2;
    -webkit-box-shadow: inset 0 1px 0 rgba(120,200,230,0.5), 0 1px 0 rgba(0,0,0,.15);
    box-shadow: inset 0 1px 0 rgba(120,200,230,0.5), 0 1px 0 rgba(0,0,0,.15);
    color: #fff;
    text-decoration: none;
}

.button-primary:hover,
.button-primary:focus {
    background: #1e8cbe;
    border-color: #0074a2;
    -webkit-box-shadow: inset 0 1px 0 rgba(120,200,230,0.6);
    box-shadow: inset 0 1px 0 rgba(120,200,230,0.6);
    color: #fff;
}

.button-primary:focus {
    border-color: #0e3950;
    -webkit-box-shadow: inset 0 1px 0 rgba(120,200,230,0.6), 1px 1px 2px rgba(0,0,0,0.4);
    box-shadow: inset 0 1px 0 rgba(120,200,230,0.6), 1px 1px 2px rgba(0,0,0,0.4);
}

.button-primary:active {
    background: #1b7aa6;
    border-color: #005684;
    color: rgba(255,255,255,0.95);
    -webkit-box-shadow: inset 0 1px 0 rgba(0,0,0,0.1);
    box-shadow: inset 0 1px 0 rgba(0,0,0,0.1);
    vertical-align: top;
}

This has the advantage of:

1. Ensuring that even if WordPress changes the class names or styles, you keep your button styles.
2. If WordPress moves the stylesheet files, you still get the styles.
3. These are exactly what you want. The alternative is loading extra styles you don’t need.

Way #2: Don’t Do This

Not quite sure why I’m even including this, but maybe it’ll be useful for someone. The alternative is to load the WordPress button styles on the front end. This has all the disadvantages that are opposite of the advantages of Way #1 listed above.

To do this would require two things:
1. Load the stylesheet.
1. Change your markup as all the styles only work if they’re wrapped in an element with the class wp-core-ui. (Which should tell you that loading these styles are not intended and a bad idea!

To load the styles, I’d do this in a child theme’s functions.php file, a custom plugin, or an mu-plugin:

function wpse159952_load_wp_button_styles() {
    wp_enqueue_style( 'wpse159952_wp_button_styles', includes_url() . 'css/buttons.css', array(), get_bloginfo( 'version' ) );
}
add_action( 'wp_enqueue_scripts', 'wpse159952_load_wp_button_styles' );