Ok. I think I found my way :)
First I went on and enabled the required stuff in sharepoint. (http://stackoverflow.com/questions/1237956/infopath-forms-services-is-not-turned-on)
Then created a simple form and published it as a web based form.
Reading this helped: http://office.microsoft.com/en-us/infopath-help/introduction-to-browser-compatible-form-templates-HA010169339.aspx
Now I'm reading this: http://office.microsoft.com/en-us/infopath-help/infopath-2007-features-that-are-unavailable-in-infopath-forms-services-HA010210587.aspx
Hopefully after that I will know better and maybe even start porting the not-web-based forms.
Thanks,
Michael