How can I improve my skills in software testing?
There are plenty of books and resources online. I have found Wikipedia is apt for covering things at high levels (search for "Software Testing" and read every referenced page). Beyond that nothing compares to real world experience.
Is there any game or something similar that I can learn which in turn can improve my ability in testing?
I haven't found anything like this (there are 'games' on GitHub to test your Git knowledge, for example). I would say try to cheat at games by looking for and exploiting bugs. Like how people "dupe" items in online games. Really, the QA mindset is "how can I change the inputs or processes to come up with a result the dev's didn't think about".