Does it makes sense to buy a new cycle to learn riding? I am a complete newbie in the cycling world but I want to learn to cycle and use it to explore the city and the places yonder. I also intend to use it to ride daily to office gradually.

So my question is that should I buy a cheap/used bike to learn to cycle first, or buy the bike I like such as a Trek DS or similar?

I ask this question as I might be overlooking the wear-tear or other points before I make a hasty purchase. I am in my mid-twenties (yes, you can shame me for not knowing to cycle) and I live in a bike friendly city on the east coast of the United States.
I learnt to ride in my 20s. I couldn't find a big enough bike second hand, except by spending several times the price of a cheap new bike.  So I bought a cheap new bike and learnt to ride on it (a steel so-called mountain bike but really a hybrid with knbbly tyres).  Even that was actually only just big enough.  I didn't really get confident in bike handling until I bought a much nicer hybrid that fitted me properly.

If you can possibly get a decent hybrid or hardtail mountain bike second hand that fits you that's the way to go.  This will help you find what features you want, as well as being something to learn on; you may even find that this is a good commuter long-termn and you get something nicer/more specialised for having fun.

There's no shame in learning to ride as an adult, in fact quite the opposite - good for you.