Similar to Brian M. Scott's suggestion, there are many tricks. The relative usefulness depends on what calculations you want to do. Are you specialized to integer calculations with exact results? In that case some of the below won't help.
The more arithmetic facts you know, the more likely you can find one to help. I find knowing all the perfect powers up to 1000, powers of 2 up to 2^16, factorizations of numbers up to 100 (especially which are prime), factorials to 8,and divisibility tests useful. Great facility with $a^2-b^2=(a+b)(a-b)$ is essential. Common trig values and Pythagorean triangles come in handy. When doing multidigit multiplies I find it easier to start from the most significant digits. You can quit when you have enough accuracy.
Approximations: $(1+x)^n \approx 1+nx$ for $x \ll 1$ is the biggest hitter. Depending on the accuracy you need, $\pi = \sqrt {10}$ and both might equal $3$. $e\approx 3(1-0.1)$ which can feed into the $(1+x)^n$