What parts do I need to change to convert from flat handle bar to drop bar? I am thinking about getting a drop bar for my Merida Speeder 200 (2015). But I am told that I need to change some other parts as well. 

My bike has V type brakes, 3 speed front derailleur and a 9 speed Tiagra read derailleur. 

Which shifter-brake lever(I mean road bike type shifter) is compatible with my bike I don't know. Also do I need a modification on brake system? Do I need to change any other parts? Most importantly, what would the cost be?

Can you help me?
To do it best, You will need the bar (obviously) and the brifter(brake/shifter) and to switch to cantilever brakes. The latter is avoidable if you use what's called a travel agent but if you have the money the canti's are the way to go, as the travel agents are ugly and I have heard a lot of people having issues with them. 

The reason for this is that V brakes have a different pull ratio than road brakes and with v brakes and a road lever you don't get proper leverage and the result is poor braking.

Look up 9sp brifter on google and you should have a plethora of choices. You should be able to use your existing stem as long as the clamp diameter is the same. 

If I had to guess your looking at a 200 bucks if your were to go through a shop, total guess on my part though, and that number could vary greatly dependent on the components and quality/brands you go with.