If I want to hydrate myself, will drinking 1 cup of black tea help or hurt my goal, on balance? What about coffee?

Varies. Tea is a diuretic -- it will make you lose water. Drink lots of water -- think liters, not cups, to stay well hydrated. Tea is ok, but you need water. An easy rough assessment is evaluating your urine. If it is clear or light colored (you can see through it) you are well hydrated. If it is dark, has a strong smell, or if you hardly go during the day, then you are dehydrated. Keep drinking.
H2O. If you drink large amounts of black tea the caffeine can lead to some degree of dehydration. Drinking smaller amounts of black tea should not cause dehydration. Caffienated coffee can also lead to decreases in hydration. However, why not drink water to improve hydration. If you are going to drink tea - green tea has more health benefits than black tea.