Hi Ren,
1. Are you trying to create the template a store it online or save it as a file?
By default, the tool will try to upload to an online account.
If you switch that to 'Save template to file' it defaults to C:\Users\<username>\Documents\ArcGIS\ProjectTemplates
2. Where is the vector and raster data stored that you're packaging up and how is it referenced in the map (letter drive, UNC path?)?
3. Are you check the 'Share outside of organization' option? Depending on how the data is referenced and whether you check this option, the size of what is being packaged could vary tremendously.
The Project Package help explains how data is consolidated depending on the above Project package—ArcGIS Pro | ArcGIS Desktop
4. If you wanted to, you could create a map package Map package—ArcGIS Pro | ArcGIS Desktop though since this is just to start new projects, that probably isn't necessary.
If you have a project with the data that you always use already in a map, you could just open that project, Save As a new one and go from there. Though I'm interested in answers to 1, 2, and 3 to see if we can figure out what the delay is.
Thanks