I am new to using ArcMap. I have a list of locations with lat/long. I'm wondering if it is possible for ArcGIS to compute the sum of the locations by state and then shade in the state depending on the total amount of locations.
You should look into a Spatial Join. There are details here from ESRI. If you follow the steps of to spatial join the locations with the state polygons, you will get a new state polygon with a count field (Join_Count) that will tell you how many points were in that polygon.
This is a great tool to get to know in GIS since it can be useful in many different situations.
Edit You can shade then based off of the join count in the new layer. You can set your symbology color to ramp in whatever shade you'd like.
What you could do is:
Do an intersect of all the points with the State Feature Class, which will get the state name in the point feature class.
Do Frequency Statistics on the point feature class with the State name as the Case field - this will give you a table with the count.
Do a join of the frequency table to the State Feature Feature class and symbolize on it.