I am looking to customise the footer links that I have set up usingthe K2 module tool.

Do I make the changes in the module and add cusom css? Sorry clueless.

I would like to remove the spacing and the bullter points. Should I be using another extension are changing css code directly?

Footer Link below shows the current layout.'

I have looked around to no avail.

It seems that this class is responsible for the bullets ul {list-style:disc;} you can override it by adding a list-style:none; to the div.k2CategoriesListBlock ul li class in your template's css file.