How does one edit and save clips once and reuse in multiple videos with Adobe Premiere?
Within a new project, create a sequence/timeline; call it 'Basic Edits' if you like, and make your trim edits on the source clips there. For each of your multiple videos, create a new sequence and copy-paste the trimmed clips from the first sequence.

Or if you want to use these clips in another project, you can import the sequence by the File -> Import menu dialog.

Of course, I presume that by Adobe Premiere, you don't mean Premiere Elements - the lite version - which doesn't feature multiple sequences in a project, as far as I know.