I'm not sure how I will start it; I tried doing some research and I don't think Swing timers fit.
But here's really what I'm after:
For example, the player chooses Timed mode, then we'll suppose there'll be 1 minute for the player to play the game, and once it's over, then scores will be recorded. (When paused, the timer will also pause).
I'm not exactly asking for codes, I'm a beginner and would appreciate a few suggestions like what stuff should I learn about, something like that to begin this task.
I'm assuming your game is event-driven, and doesn't use a main program loop (usually the case with real-time games).
When your player starts their turn, you can pass a TimerTask to java.util.Timer that executes once or twice per second (use scheduleAtFixedRate()). Each time it executes, you update the GUI (just call repaint() and let the GUI decide how many seconds are left). When the TimerTask determines that time has run out, it tells your model that the turn should end, and then it performs cancel() on itself.
There are two sorts of Timer, which work slightly differently. You can read about the other one (Swing) here. Then you can decide which sort is better for your current requirements. If the Swing Timer works in the Event Dispatch Thread (EDT), you may find it blocks your entire GUI, in which case you will want the other Timer. In Timer, it suggests this link will help; I haven't read it.
Remember you need to enter the EDT so your Timer or TimerTask would have to enter it: start reading here.
Actually, there is a third kind of Timer (Management) in the API.