Q: deleteing tunes

If you want to manually remove songs on an iPod (instead of using automatic syncing), there's a setting to Manually manage music.  Connect iPod to iTunes.  The screenshot below shows upper left corner of my iTunes window, showing the iPod's device button (circle in red).

Click device button to show iPod's Summary settings screen in iTunes.  On that screen, checkmark box for Manually manage music.  Click Apply.  Along left side of window, in the sidebar, under On My Device, click Music.  To the right, iPod's current song list appears.  Select the song you want to remove from iPod.  Press Delete on keyboard.  You can do this with more than one song selected at the same time.

Is your iPod a 1st or 2nd gen (older type) iPod shuffle.  If so, please post back with model.  They work differently from the "big" iPods and later shuffles.  This document shows the four iPod shuffle models

Identify your iPod model - Apple Support

4th gen shuffle works as described previously.  Are you able to click the iPod's device button as described in my first reply?  If so, the iTunes window looks like this

This is the shuffle's Summary settings screen.

To the right, the setting (checkbox) for Manually manage music is circled in red.  Checkmark box (if it's not already) and Apply.

Then, in the sidebar, under On My Device, click Music (circled in red).  The shuffle's song list appears to the right.  Select song (that you want to delete) on list.  Press Delete on keyboard (or right-click song and Delete).

If that's not what you see, please describe what you see (or include a screenshot).