Google index new websites

Yes, Google does it automatically. If it's about your site, you can do a few things like creating a sitemap or internal linking.

Your website should be search engine optimized in order for Google to imdex your website. You can also ping your website should you want to speed up the indexing process

Google WILL NOT automatically index your website if it does not know it exists. How could they know?

The best way to get your site noticed by Google is by setting up an account in Google Search Console. You should also do that with Bing Webmaster Tools.

Aside from that, the only way they will eventually discover your website is from other websites linking to it. Even a link to your website on Facebook or Twitter should eventually get picked up by Google.

Listing your site, verifying your domain, and submitting your site's sitemap through Google Search Console will do the trick to get your site indexed in Google.

Index your site with ping ,create a sitemap ,register your site to google analytics
This will make google more easy to index your site

Thanks guys, it's actually a lot faster when i declare everything on Google Search Console

Assuming you're using Wordpress then install an SEO plugin like RankMath or Yoast and use their 1 click sitemap generation feature then submit those sitemaps to Google Search Console and Bing Webmaster Tools.

If you pay for the premium version of RankMath you can enable "Instant Indexing" and it'll attempt to index your posts automatically for you.

Both GSC and BWT also have options to manually submit your links to their search engines. Bing allows you to mass submit all URLs in one go and Google URL by URL but with a small limit of how many you can submit in a 24hr period.

Aside from that to help speed up the discoverability of your site from Search Engines setup profiles on all the major social media networks, social bookmarketing sites, web 2.0 sites, directories etc and post your website link and / or content - these links won't help you rank in Google (they carry little weight because they are so easy to get) but they can help Google discover and crawl your website faster and therefore make it more likely to be indexed.

Failing that there's various indexing services, none of which I've used personally, but I've constantly heard that IndexMeNow is the best and most reliable.