Do we have to take a breath when we start after we stop in Arabic, or while reading Hadiths or Quran?
Wa alikom asslam wa rahmatu Allahi taala wa barakatuh,
First of all, when reading hadith, you don’t have to recite it or apply the rules of tajweed. You may certainly apply it, but it is not necessary.
As for stopping when reciting the Quran, yes, you should take a breath when you make a stop whether the stop is caused by a stopping sign, end of an aya or any other reason. Just make sure to keep the rules of stopping in mind, check this lesson for more information: https://www.youtube.com/watch?v=j3AR6-BThPU
You can also make use of this tajweed infographic: https://arabic101.org/product/stopping-resuming-in-the-holy-quran-a4-poster/
The only case when stopping requires that you do NOT take a breath is at this sign س which I explained in this lesson as well: https://www.youtube.com/watch?v=meQsEM3V2m8
Hope this answers the question in shaa Allah.
Allah knows best.