jmc

Obviously, there are $6$ one-topping pizzas.

Now we count the two-topping pizzas. There are $6$ options for the first topping and $5$ options left for the second topping for a preliminary count of $6\cdot 5=30$ options. However, the order in which we put the toppings on doesn't matter, so we've counted each combination twice, which means there are really only $\frac{6\cdot 5}{2}=15$ two-topping pizzas.

Adding our answers, we see that there are $6+15=\boxed{21}$ possible pizzas with one or two toppings.