How does domestic demand affect the domestic currency's value If an economy grows, due to a tax reduction for example, how does that affect the exchange rate? 

My logic would be that reduced taxes put more money into people's pockets, which they then use to buy more stuff. If all of that stuff were produced domestically, nothing were to happen to the exchange rate. If some of that stuff were to be produced in other countries and imported, those importing the goods will want to exchange the domestic currency for their foreign currency. That would increase demand for foreign currency relative to domestic currency and thereby decrease the value of the domestic currency.

However, as the economy grows other country's producers will want to buy into the economy's market, for which they will need that country's currency and increase demand for it. That will drive the currency up. 

Therefore: What effect is stronger in response to domestically motivated economic growth? Is it the depreciating effect of imports or the appreciating effect of new entrants or am I missing something entirely?
What effect is stronger in response to domestically motivated economic
  growth? Is it the depreciating effect of imports or the appreciating
  effect of new entrants or am I missing something entirely?


There's no general answer to that. It depends on the particulars of each economy. Some factors are:


Whether imports outweigh exports or viceversa;
Whether the tax reduction is coupled with spending cuts, tariffs,
foreign debt, etc.;
Taxpayers' general propensity to save the additional "disposable"
income could influence the exchange rate in either direction,
depending on where they deposit their savings.