A tax imposed by the government on the sellers would drive up the price of the good. I understand that the supply curve will shift upwards due to this but the won't the demand curve also shift downwards due to the increase in price of the good?
I think you're confusing quantity demanded, and the demand curve.
The demand curve tells you how much quantity should be demanded at each price point. Therefore changes in price should change the quantity demanded, not shift the curve. You want to think of this in terms of demand and supply shocks.
Imagine it was discovered that bananas gave you super powers. Now the ability to supply bananas hasn't changed (no supply shock), but the desirability of bananas at any given price point has increased (demand shock).
An increase in price due to taxation has not impacted the desirability of goods at any given price point. It simply changes the price point.
The demand curve is a function that shows the relationship between price and quantity demanded. This curve is made up of an infinite number of points, with each point corresponding to two pieces of information: a price and the quantity that would be demanded at that price. If you consider a different price, you only need to then consider a different point on this curve - so there would be a movement along the curve if the price changes. We say that price is endogenous to the model.
When the demand curve was constructed, some other factors were assumed to be constant. These are the factors that do not appear directly on the graph (like price), but are behind the scenes. So, for example, if you want to consider how government expenditure affects a demand curve, you would need to draw a different demand curve based on the different level of G, because G is assumed to be constant for any one particular demand curve. We say that these other factors are exogenous to the model.
The reason why taxes on production shift the supply curve upwards is because they increase the cost of production. Those costs need to be added onto the price of the product, so the price increases for any particular quantity.