Does it makes sense to buy a new cycle to learn riding?
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.