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I put 4 different layers into my ArcMap document, but only one of them will show up on the map! I should be seeing the other 3 layers as well. When I do a Zoom to layer, I can see one layer, but I cannot figure out how to make the other 3 show up at once on top of each other.

Before I opened ArcMap, I used ArcCatalog where I right-clicked 3/4 of the shapefiles and clicked Properties → Coordinate system → Import and then clicked on my Trees layer, which has the coordinate system I want the other 3 to have. But when I opened ArcMap the 4 layers did not overlap, nor could I move them above or below each other.

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You are confusing two different operations dealing with Projections.

  1. Define a projection: You would use this when you have a shapefile or other feature class that does not currently have a spatial reference defined when you look at the properties in ArcGIS. You might also use this if you knew based on some observation, that the projection defined for the layer was wrong. This could be due to recognizing that the coordinates of the features were not appropriate for the currently defined projection, or some other clue. This operation simply overwrites the existing projection with a different one that you specify. This can also be done in ArcCatalog by changing the coordinate system in the "XY Coordinate System" tab of the feature class Properties.

  2. Project a featureclass: This tool is used when you have a featureclass with a defined projection, and you want to convert it to a different projection. This will run the appropriate transformation to change the features to the coordinate space of the new projection.

In your question, you state that you have a shapefile that is currently in a Geographic coordinate system or Latitude/Longtude which has units of Decimal Degrees. This means that you have coordinates in the range of Y/Lat = 0 to 90 and X/Lon = -180 - +180. When you bring this shapefile into ArcMap untouched, its projection will be recognized. If you use the measure tool set to miles, ArcMap will calculate the distance correctly.

You then used the Define projection tool, which doesn't convert the coordinates, but simply changes the projection assigned to the shapefile from Geographic to State Plane NAD 83 California Zone 6 Feet. This gives you a shapefile with units in feet. The features in the shapefile did not have their internal coordinates converted, however, so they are still in the range of X = -180 to +180 and Y = 0 to 90. Running the Project tool using the same coordinate system doesn't actually accomplish anything because you have already set your shapefile to this projection with the Define tool. This is why when you use the Measure tool, the distance is dramatically smaller. Essentially, your all the features of your shapefile are now fitting in a distance of 360 feet horizontally and 90 feet vertically.

The proper procedure, to be performed on the original shapefile in the geographic coordinate system, is to use the Project Tool. For the Projection, choose State Plane, NAD 1983 California Zone 6 Feet. The output should be in the proper location and give a correct distance when measured. It should also overlay correctly on any other layers, provided they have a projection defined and that your Data Frame in ArcMap has a projection defined.

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Although you have set the projection of your 3 other layers by the properties menu in ArcCatalog, you have failed to re-project the data to the proper coordinate system.

Please try the Project tool, located in Data Management Tools → Projections and Transformations → Feature.

This should allow you to create a new feature class that is projected in the correct coordinate system.

If by chance your 3 shapefiles are of unknown coordinate system, please make sure to set the coordinate system (before proceeding with the above Project tool) for these shapefiles to that of their original projection by using the Define Projection tool, located in Data Management Tools → Projections and Transformations.

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This a simple tutorial one can follow to define (overwrite) or reproject (transform) a Coordinate Reference System (CRS) in ArcGIS 10.1+ starting from a CSV file. To clearly understand 'define' and 'reproject' concepts refer to this answer.

  1. Open ArcMap;

  2. Click on File --> Add Data --> Add XY Data;

  3. Browse the CSV file which contains column A with longitude (or Easting) values; and column B with latitude (or Northing) values. Example:

        easting     northing  
 732771.529  7457694.818  
 733006.267  7456350.588  
 735506.391  7481351.041

3.1. Select column A as the X field and column B as the Y field from the CSV file;

3.2. On Coordinate System of Input Coordinates field, select current CRS from data (or choose one manually).

Here the CRS is defined. One can export the layer to a shapefile by right clicking in the layer and then, Data --> Export Data.

Or continue (from item 3.2) to reproject:

  1. Click on ArcToolBox button (little red toolbox icon), then go through DataManagementTools --> Projections and Transformations --> Feature --> Project;

4.1. Select the input layer from the Input Dataset or Feature Class field;

4.2. In Output Coordinate System choose the CRS one wants data to be projected (transformed) to.

4.3. Set file path and name for the output file in Output Dataset or Feature Class field.

4.4. Click Ok and wait while Project finish running.

Here data is already reprojected to new CRS.

Now, let's create columns with new X and Y coordinates in the layer's table of attributes:

  1. Right click the new created layer and select Open Attribute Table;

5.1. Click on Table Options (upper left corner), then Add field to add new X coordinates;

5.2. Chose a name for the column (e.g. X_coords) and set the Type field as Double.

5.3. Repeat steps (5 to 5.2) for creating column Y_coords;

  1. Go back to the Attribute Table and right click X_coords;

6.1. Select Calculate Geometry. Set Property for X_coords equal X coordinate of Point. Verify if Use coordinate system of the data source is selected. Click Ok.

6.2. Repeat steps 6 and 6.1 for column Y_coords, but select Y coordinate of Point instead.

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