Wordpress - Woocommerce get cart total price in a number format

That is what you want:

Working with global variable:

global $woocommerce;  
$woocommerce->cart->total;

Working with function:

WC()->cart->total;

Update 2020

Answer

See flytech's answer for a solution using the native WooCommerce API.

Note / Caveat

If you're going to do proper arithmetic with monetary values, always use signed integers (!) representing the smallest denomination of a given currency (Cent, Penny, Paisa, Dirham, e.g.).
Only convert back to decimal fractions in the presentation layer of your application after all calculations are done.

This holds true regardless of language or framework.

Original answer

I don't know woocommerce at all and hence there might be a native way as well, but anyhow, this

$amount = floatval( preg_replace( '#[^\d.]#', '', $woocommerce->cart->get_cart_total() ) );

should do.

The preg_replace eliminates everything but decimal characters and colons.

Should you care to do math with it, the floatval converts the value from a string to a numeric one.