memorizing squares, cubes, etc.

I agree with bmw. Athough to memorize cubes till 10 is a bit too much for me.

I use a trick, if you can call it that way, which I find particularly helpful.

If you want to calculate something like 19^2, you can think as follows:

19*19 = (20-1)*(20-1) = (20-1)^2 = 400-40+1 = 361

Same holds if you want to calculate products like 23*23. To transform in into (20+3)^2 = 400+120+9=529 works in less than 10 secs, especially if you are good in numbers.

Last but not least, there is another trick I use which is extremely helpful.

If you encounter products in the form 999*1001, or 19*21, use (a+b)(a-b)

19*21 = (20-1)*(20+1) = 20^2-1 = 399
999*1001 = (1000-1)*(1000+1) = 10^6 -1 = 999999

Hope this helps.