So… can I add a completely normal plain form to my hugo site?
Yep you can have a normal form on your template, but how do you make it sending the information, via POST to an email for instance?
You need some sort of server code to do that, and the easiest way to do that on a static site is to rely on third-party services like Netlify (if you host the site there) or others like https://formspree.io/ or https://statickit.com/
Just tell me then how I make a form using markdown or something and I figure out the rest
You can write HTML in Markdown