I am trying to find the best route and the shortest driving time from districts to a highway. My guess is to use Network Analyst for finding the best route. However, I wasn't able to even begin by creating a new route layer. The option was grayed out. Did anyone know how to solve this? Also, can I also use Network Analyst to find the shortest driving time? I am very new to ArcGIS so any suggestions will be really helpful! Thanks!
You didn't say whether you're using ArcMap or ArcGIS Pro, but I'm guessing ArcMap since you're describing grayed-out behavior. The option to create new Network Analyst layers in the ArcMap toolbar will be grayed out if you have not added a network dataset to the map. A network dataset encompasses routable streets and various settings for how travel time and distance can be calculated.
You should probably start with the Network Analyst tutorials here: http://desktop.arcgis.com/en/arcmap/latest/extensions/network-analyst/about-the-network-analyst-tuto...
Or, for ArcGIS Pro, start here: https://pro.arcgis.com/en/pro-app/help/analysis/networks/route-tutorial.htm
Note that if you don't have a network dataset and don't wish to create or purchase one, you can use the ArcGIS Online routing service, which instead consumes credits for each analysis you perform. This is much easier to use from ArcGIS Pro.