Gravitational Field Strength in cities 
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*How can we experimentally measure the gravitational field strength of a place? 

*Which cities have the lowest / highest gravitational field strength?
 A: 
How can we calculate the gravitational field strength of a place ?

I'm assuming you meant the local acceleration due to gravity g rather than field strength $\Phi$. There's a big difference between the two. Field strength has units of energy per unit mass, and is nearly constant at sea level if one incorporates centrifugal potential. Acceleration due to gravity varies over the globe due to centrifugal acceleration and the Earth's equatorial bulge (which is also caused by the Earth's rotation).
With regard to g, the two largest factors are latitude and altitude. A lesser factor is the nature of the rock underfoot. Ignoring that final factor, gravitational acceleration (including centrifugal) at sea level can be calculated via the Somigliana gravity formula $g = g_{\text{eq}}(1+\kappa \sin^2\lambda)/\sqrt{1-e^2\sin^2 \lambda}$, where


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*$g_{\text{eq}} = 9.7803267714\,\text{m}/\text{s}^2$ is gravitation at sea level at the equator.

*$\kappa = 0.00193185138639$, reflecting the observed difference between gravitation at the equator versus the poles.

*$e^2 = 0.00669437999013$ is the square of the eccentricity of the figure of the Earth.

*$\lambda$ is the geodetic latitude of the point in question.
Altitude also has an effect. Gravitation on the surface of the Earth decreases with altitude by about $1.967\times10^{-6}\,\text{m}/\text{s}^2$ for every meter of elevation above sea level. This is the Bouguer correction. It assumes a large slab of rock of uniform height and of average density, 2.67 grams per cubic centimeter.

Which cities have the lowest / highest gravitational field strength?

Equatorial cities at high altitude have the lowest gravitational acceleration, while northern cities near sea level have the largest. Quito, Ecuador sits right on the equator and is at an elevation of 2850 meters above sea level. Murmansk, Russia is at 68.95 north latitude and is at sea level. Using the above expressions, gravitation is about 9.775 m/s2 in Quito and 9.825 m/s2 in Murmansk. Both Quito and Murmansk qualify as cities. There are towns, villages, and scientific outposts that are near the equator and higher than Quito, or that are near sea level and further north than Murmansk. 
