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: 
Instead of a single "coming soon" or "under construction" line, expand it a little and explain *what* is coming soon and why a visitor should bookmark it or visit it again. 

There's also no reason why you can't already include a short about paragraph, some contact information and possible even a little subscribe function where people could enter their e-mail address to be notified when the site is done. Other than that, don't put in any unnecessary redirects or content that wouldn't be relevant to what would eventually be available on the site.
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.