You're running the analysis multiple times, I assume? If you only need to run it once, saving the output is your best bet. (When it gives you the output, which will be in a different window from your data file, click file and save. You should be able to save the .spo file to open later with SPSS).
If you really are running the same or similar analyses over and over (e.g., with slightly different data sets, etc), then the best way to save this analysis is to use syntax. Instead of clicking OK before you run your t-tests, click 'Paste'. This will open a new window with the code that runs the t-test. Save that syntax, and you can open and run it any time in the future.
I hope this helps!