Is it possible to install new template and adjust website to it, without anyone else seeing changes until i decide to go public?
Yes there are ways you can do this.
Specify Template in URL Parameter
You can install the new template and specify the new template via a parameter in the url like this: https://website.com/index.php?template=newtemplatename
Test Pages
You can create test menu options and assign these menu options to the new template in the Menu Assignment tab in the template settings.
Alternatively Set Up a Test Copy
Unfortunately, the above testing methods may not work very well as a new template will likely have different module positions and names so your modules may not display as you would expect.
For this reason, it is often easiest to make a test copy of your website where you can change the module position names to suit the new template for a complete test.
It can be fairly straightforward making a copy of a Joomla website using the free or paid version of Akeeba Backup and Akeeba Kickstart and placing the new website in a sub directory such as /test or similar without interfering with the live website.