Can anybody help me with a solution to remove the shipping price in the checkout process, and also in the order email. Right now it is set to 0,00 but I want to remove the price, so only the delivery text is displayed.
All you need to do to solve this is to rewrite the shipping render total. That is done pretty easily and lets you make changes whenever needed.
You may use your current customization module, or create a new one from scratch - here you will need the rendering block (instead of the default one):
Here we change only the template, so you can inherit it from the default one and rewrite only the parts you need. Alternatively you can create a new one. It's up to you to decide and depends on your requirements for the block.
Next, you will need the template mentioned in this block:
You can get it by copying the default (app/design/frontend/base/default/template/checkout/total/default.phtml for the basic theme).
Right in the template you only need to hide the price (using the Hidden tag) or remove it.
And finally you only need to modify the layout of the objects you need to use for the directives. For the default shopping cart and checkout it will look like this:
And this is what you should get: https:www.//gyazo.com/b3208c432a574a35303738e4119ffab5