I purchased a domain and used it to build for fun a website for a school dance. Now I removed it and replaced it with my actual website with a totally different topic (piano lessons). While looking on Google search results with site:pianogenius.it I still see some pages of the old website (that doesn't exist any longer) indexed that lead to 404 Is that bad for SEO? Should I purchase a new domain, or is just a matter of time that Google will forget the old pages and will index the new ones (that hasn't already done)?
It takes time for search engines to properly index your websites. Changing the entire website within a 3 month timespan will impact your SEO for sure. Search engines love it when websites are easily identifiable. If your site looks like it's going through a mid-life crisis, it's not going to rank well.
If you need more SEO related help feel free to message me.
I do quite a bit of site migrations (rebranding, domain changes, redesigns). Usually the goal is to preserve (and improve) organic traffic as well as preserve page authority of old pages (and transfer it to new pages).
I see that your case is different because your two sites are not related. And your first site hasn't had a chance to aggregate links (authority) yet.
The short answer is that your current situation should not affect SEO of the new site. "404 or 410 Status Codes Will Not Impact a Website’s Rankings: If Google identifies 404 or 410 pages on a site, it will continue to crawl these pages in case anything changes, but will begin to phase out the crawling frequency to concentrate more on the pages which return 200 status codes." Source: https://www.deepcrawl.com/knowledge/hangout-library/4xx-errors/
However, here's few things you can try to simplify Google's life in understanding your site:
1. Verify your domain with Google Search Console (GSC)
2. Create and XML sitemap and submit in GSC
Although "site:" is a good quick lookup option, Google suggests to use Coverage report in GSC to get more precise and reliable information on the state of your page indexing. Keep an eye on that report and don't stress too much about 404s in SERPs. They'll get removed eventually. It might take up to 12 months for Google to remove those completely but in most cases it happens much faster.
I hope it helps. Feel free to book a call if you have more questions.
School dance and piano lessons are two totally different poles. I'm not sure how you related the two as same.
However, if your domain has the word piano in it, it's okay to create a website around piano lessons as you have already mentioned.
Regarding the indexed links, you can resubmit the new sitemap in Google search console and inform the search engine about it. This way, your previously indexed pages will be deindexed in a day or two.
Hope this helps.
Suppose I published an article covering "BEST SEO PRACTICE OF 2020", at that point in a month or something like that, I overhaul, include or update the article, which really is designated "Curating content" which is a very healthy approach. But if I keep the URL, and then change the entire content and the topic, then it would be not helping my SEO at all. Remember that the content should match your topic, title tag, and URL. You can not put some random content on a URL containing some other keyword!
What you can do is to REDIRECT OLD URL to the NEW one! which would still be a good idea!