Question: I am setting up my personal website, and I have designed a simple "Coming Soon / Under Construction" page which basically has nothing but a small notice for the users. 

What kind of SEO can I make on this page such that it won't harm the full website's ranking later on?

Answer: 
Make sure the content/keywords on the coming soon page is similar to the content that your final website will contain. Newer websites tend to get crawled less frequently so it may take a while to get your new website crawled and Google's index updated. So the snippet Google displays in their index may not be updated for a while after you launch your new site. You want to make sure it still jives with what your new sites has to offer.
A good place to start would be to create a simple homepage with an email registration form for 'Early Bird Access'. Add a blog article (highlighting the future launch of your project) and some legal pages (e.g. Terms and Conditions, Privacy Policy, and Cookie Statement).

If you've got Google Webmaster Tools setup and ready to go, create your sitemap.xml and submit your pages for indexing - they can show up in search results within minutes, not weeks.

It's important to be able to find your website, product or software without having to use the full domain as the Google search terms (e.g. www.example.com), so get indexed in Google as early as possible, even if that means you only target the name of your software and not the domain (e.g. example) - as some people will simply type 'example' rather than 'example.com', and you'll want your website to appear at the top of the list.