Help with water leak

My house is on a slab and has city water. I noticed my last water bill was high and I think I hear a "water running" noise coming from the walls or ceiling of my downstairs interior bathroom. So this made me think I have a leak. I checked the water meter and sure enough the triangle is spinning like crazy. I shut off the main water valve to my house which is in the garage and the valve to the irrigation system which is in the ground next to the meter. This did not stop the meter from spinning.

Questions:
So I guess this means the leak in between the meter and my main water valve?

Who should I call about this, the city (I am on city water) or a plumber?

Do you think the noise I hear in the bathroom is related to the leak? It still makes the noise when the main valve to the house is closed?

Also whats your guess at a cost to fix an underground leak between the meter and the main valve? $500, $1000, $2000.... I have no idea.

Thanks.
Saint.

If you shut off the main valve and the meter is still spinning then the valve does not hold. Usually on a city meter there is a shutoff on each side of the meter, one on the street(city) side the other on house side. The first thing to do would be to install or replace the main valves, wether you or a plumber does it. When the plumber is there then have him investigate you're leak, he/she maybe able to find other valves or add valves to isolate the leak.

If the problem is on "your" side of the meter, it's your problem. If the meter still spins with everything turned off, you have a serious leak, likely the main supply line between the meter and the house. How much to fix? No way to answer that, we don't know what or where the problem is for sure. If it is the supply line, it will require digging a trench to install a new line, the cost will depend on how much digging has to be done. If it is the supply line, the major cost will be labor for doing the digging. Materials and the actual plumbing cost will not that much. My advice is to get a qualified plumber to evaluate the problem and go from there, and do it now. If a new supply line is in your future, you could save money by doing the digging yourself or by hiring someone looking for work.