Shifter clicks in but chain stays on the largest cog, when riding I am able to change (click through) the shifter but the chain stays on the largest cog and does not move (This happens only when riding) but the gears shift as usual when the bike is stationary and I turn the pedals with my hand.

The bike is new, about a month old, everything worked well until last week and no, I'm not able to push the derailleur with my hand. I had a problem before when I was not able to shift (shifters won't click into the 3rd) into the 3rd gear (the largest cog), this was corrected when I took it to a bike mechanic. Now (about 4 days later) I've got a problem shifting between 2nd and the 3rd. Sometimes it shifts and most of the time it doesn't (shifting into and out of the 1st works fine). Btw I lubricated the chain after I got back from the bike mechanic.
Given its only a month old - return to the shop for the first "tune-up" which is standard in most parts of the world.

If you bought the bike on line, you can either learn to do it yourself or take it a bike shop and pay for a tuneup.  Depends on your mechanical aptitude.

Sounds like your bike mechanic visit missed something.