What easy algorithms are there for calculating products of cycle decompositions?
Here is how I would do it manually, taking the example of $(123) \cdot (12)(34)$.

Start with a number appearing on the right permutation, say 1. The right permutation sends it to 2, and the left to 3.

Doing the same with 3, we get 4 and then 4.

Doing the same with 4, we get 3 then 1. We have closed a cycle $(134)$.

Now take another number on the right (or left) permutation not already covered – has to be 2. It gets sent to 1 then 2. We have closed a cycle – 2 is a fixed point.

In total, the result is $(134)$.

Now let's compute $(12)(34) \cdot (123)$.

Let's start with 1. It goes to 2 then 1. So it's a fixed point.

Let's try 2 next. It goes to 3 then 4, which goes to itself then 3, which goes to 1 then 2. We have closed a cycle $(243)$. That's the answer.