Tableau Filled Maps at the City Level

I am working on a filled map showing in gradual colors visitors from each state, but how do I get the map down to more granular scale than just state? Whenever I try to use the city dimension, Tableau pops a warning message saying that the city field can't be used to create the filled maps . Does this mean we need zip code level or some other dimension added in there as well? I've already tried all of the different globe options. 

You can't use City in a filled map because Tableau doesn't have the borders for each city. It does have the lat/long though, so just change your chart type to Circle. Also, adding Country and State to the Detail shelf will help Tableau identify more of the cities. Otherwise you get a lot of Ambiguous city locations.

OK, but once I convert the geo map to circles, there is no relative depth to the shape in terms of its measure (a.k.a one city which has a lot more website sessions doesn't have a different sized/ colored circle than a city with significantly less sessions). Is there an obvious way to make these differences more apparent on this level of detail?

You're right. Although you could add sessions to the color and/or size shelves, maps are best at showing where things happen, not so much their intensity or frequency. You normally end up with a few outliers that throw off the range of values and makes it hard to see most differences. I recommend a combination of charts - perhaps a map at the top and a bar at the bottom. They don't have to be at the same level of detail - you should use actions to filter or highlight selections.