When everyone sells at an event of a recession, who buys? I'm trying to walk through and understand the basic events of the great depression. I understand that at one point (Black Thursday) the news of a downfall of stock prices hit investors, so they obviously wanted to get rid of their shares.

Given the very high pace investors started to sell, there needs to be an equally big number of buyers, or -- more likely -- a not that big number of buyers, who bought that huge number of shares.

Is this a valid assumption?
If it is, I don't see who bought those shares.
the news of a downfall of stock prices hit investors, so they obviously
  wanted to get rid of their shares.


This is not the way you want to think about prices. Stock prices are determined via the equilibrium of demand and supply. It sounds like you are assuming that they drop from the sky and investors react, but they are determined via the actions of investors. What happens is that news of the fundamentals of the economy (Things like interest rates, employment numbers, job forecasts, etc.) hit, investors react, and then price falls. 

Suppose there are two investors A and B and a single stock. Suppose that $t=0$ both investors 'subjectively value' the stock at 10, this valuation is determined via each investors subjective expectations of future earnings and future prices. So the price at $t=0$ is 10 and lets suppose that investor A owns the stock. At $t=1$ news about the fundamentals of the economy hits, each agent updates their subjective value of the stock. Suppose that the 'value' to agent A is 5 and to agent B is 6. Its clear that Agent A will trade the stock to agent B at a price between 5 and 6, hence price falls endogenously not exogenously.  As this example illustrates, trade occurs between pessimistic sellers and relatively optimistic buyers.