I've no doubt you are right - computing skills may be acquired and taught to elementary school children. Unfortunately, I am not aware of any Internet site that provides specific information. But here a few books (certainly there's more) that cover various aspects of the subject.
Two things I am absolutely certain of is that (1) there is no theory unifying all the possible tricks, and (2) most of the tricks depend on your memorizing some basic facts.
For example, if you remember that 12 squared is 144 you'll be able to assert that 11*13=143 right away. Why? Because (n-1)*(n+1)=n2-1.
Another example. What's 452? 4*5=20 to which you append 25 to get 2025. Why? Because, (10a+5)2=100a2+100a+25=100*a(a+1)+25.
By the way, 11*13 is the case of multiplication by 11. This can be done by pulling the digits 1 and 3 away from each other to give room for another digit which is computed as the sum 4=1+3. For example, 11*45=495 for 9=4+5.
There are many more like these. Should you find something interesting or surprising please let me know. I've started a Fast Arithemtic Tips page that I hope will grow with time.