Sahih Bukhari and Sahih Muslim are what we taught as Muslims to believe and trust. They are reliable sources of Hadith, not because we have been taught that in schools and Mosques, but because they have strong Sanad in every Hadith that is stated in their books.
As for Musnad Ahmad and Tafsir ibn Jarir at-Tabari, if they contain weak hadiths, then you better skip them. If you encounter a Hadith that you are unsure of its reliability, then the standard procedure is to look at its Sanad and follow the narrators in the ascending order of the year of their death. If a conflict happens regarding their order in the Sanad of that Hadith, then it's considered weak, and should be skipped.