Ok, So I have a wordpress installed on a test server. I just edited the default theme to get my desired result. My question is could I just install a new version of wordpress on another server and copy/paste my theme - simple as that?
If you haven't added a bunch of content and widgets on the local server, yes, it can be as easy as that.
If you want to move everything - all your settings and content and such - you can follow these instructions: http://www.billerickson.net/how-to-move-your-wordpress-website/
If the only thing you want to move is theme, than it is indeed straight copy/paste.
But I want to note that default theme gets updated and overwritten with WP updates, as well as by its own separate updates. If you want to tweak it you should create child theme for it and implement tweaks in it.
If you want to move content as well things get a little more complicated. In basic case you can use native export/import feature, for more complex cases there are plugins around for moving installation around, adjusting URL changes and such.