How to get a new personal academic webpage to show up in searches
The usual approach would be to create links to the website from other prominent indexed websites. So for example, you might want to create a link to the page from your university profile page. You could also add links from ResearchGate, LinkedIn and other social networks that allow you to share a link to a personal website. You could start creating content and sharing it on social networks.

Then, you just have to wait a bit for the web crawlers (particularly Google) to find and index you. This might take a month or more.

If you are specifically interested in being indexed by Google Scholar, check out these guidelines:


  Individual Authors
  
  If you're an individual author, it works best to simply upload your
  paper to your website, e.g., www.example.edu/~professor/jpdr2009.pdf;
  and add a link to it on your publications page, such as
  www.example.edu/~professor/publications.html. Make sure that:
  
  the full text of your paper is in a PDF file that ends with ".pdf",
  the title of the paper appears in a large font on top of the first
  page, the authors of the paper are listed right below the title on a
  separate line, and there's a bibliography section titled, e.g.,
  "References" or "Bibliography" at the end. That's it! Our search
  robots should normally find your paper and include it in Google
  Scholar within several weeks.
  
  If it doesn't work, you could either (1) read more detailed technical
  guidelines in this documentation or (2) check if your local
  institutional repository is already configured for indexing in Google
  Scholar, and upload your papers there.