You could certainly do this with the Spatial Analyst tools in ArcGIS. I've not specifically worked with groundwater data, but I've used both Surfer and ArcGIS for contour mapping. First, make sure you have and have enabled the Spatial Analyst extension in ArcGIS (Customize menu -> Extensions -> check Spatial Analyst). Then add your well dataset as a layer; I'm assuming it's a point layer of well locations with an attribute field for well water elevation/depth? Then, select one of the Interpolation algorithms under the Spatial Analyst toolbox (Spatial Analyst Tools -> Interpolation) in order to create a uniform raster dataset from the x,y,z values in your well dataset- this step is much like using the Grid -> Data tool of Surfer, and you have many of the same options for which interpolation algorithm you can choose, e.g., Inverse Distance Weighting, Kriging, Natural Neighbor, etc. Once you've created a raster dataset, you can generate contour polylines from the raster at whatever interval you choose using Spatial Analyst Tools -> Surface -> Contour. Hope that helps!
With regard to accuracy, that's going to be influenced mostly by factors other than whether you use Surfer or ArcGIS. Any time you're interpolating, you're essentially "making up" data to fill in gaps between real data points. How accurate this is depends on the nature of the phenomenon you're trying to map and the quality of your data. Does the variable you're mapping change across space smoothly and continuously? Is your dataset a good representative sample with sufficient density and coverage? You might want to research the various interpolation algorithms to determine which is best suited to your application. But otherwise, accuracy is essentially a function of what you put in: garbage in, garbage out, as they say.