After finishing salah what should be reciting next? and are you meant to do dhikr after each fard prayer?
You can do various forms of dhikr or dua. The most common dhikr, which is done after every prayer, is to say subhanAllah 33 times, alhamdulillah 33 times, and allahu akbar 33 times, mentioned in Sahih Bukhari. You can also shorten these to ten each, aslo mentioned in Sahih Bukhari.
You can also recite any other dhikr words you know if you choose.
After that, you can perform dua, or stand up to pray the voluntary salah, if there is a voluntary salah associated with the obligatory one.