How to increase traffic to your new website quickly and for free？

Build something that a lot of people are going to want or speak about

Unfortunately this is the crux of it. SEO and socials are only free if you don't value your time.

It'd also help to know what kind of website the OP runs, obviously.

1) SEO texts. Try phrasing like "alternative to {competitor}" for free-riding off your competitors back.

2) Get backlinks. Make sure other websites have links to yours. This will increase your ranking in search machines.

3) Do social media

Write things worth reading, in theory. In reality, have friends at Google.