Making a 2-Steps-Form

I want to create a small form (eg. booking form) at my startpage with 3 fields like name, date and mail address. After clicking the submit (or next step) button the visitor will be redirected to the booking page with a large form and more fields for the booking process. And the 3 filled fields are here also filled in.

example page:

https://www.check24.de/profis/umzug

How can I create these two forms? It is not the multi-step-option, this will be a form at the same page – I want to redirect to another page!