Easiest way to create measured intervals between shape points

To create a regularly spaced point grid, use Create Fishnet (Data Management). In the attached image, I created a 50x50 fishnet grid. However, you can specify any sized grid you would like--just make sure to check the "Create Label Points" box in the tool.

I should also point out that the regularly spaced point grid approach is appropriate for interpolation methods such as local polynomial interpolation that assume samples were taken on a grid. There are other interpolation methods that do not assume samples were taken on a point grid. Methods such as global polynomial interpolation and empirical bayesian kriging do not have such assumptions. In case you are new to kriging, I would highly recommend going through through the ArcGIS Geostatistical Analyst Tutorial, which is very good.

Edit:

To add points along the feature vertices, use Feature Vertices to Points after you create the fishnet (see below example).

if you can use arcpy this code could help you :

cur = arcpy.InsertCursor("ticks")

for i in range(int(extent_XMin) ,int( extent_XMax )  , int( interval) ):
    for j in range(int(extent_YMin) , int(extent_YMax) , int( interval) ):
        pt = arcpy.Point(i , j)
        row = cur.newRow()
        row.Shape = pt
        cur.insertRow(row)

this script creates a grid in "ticks"