Making code run faster

Lets say you have some "code" for example an API endpoint or some user interface that you don't think is fast enough, and you'd like to improve it. Where do you start?

I usually start by getting some numbers on how fast it is. If it is a API endpoint I try to test it more than once, noting response time. I usually re-start the service test it like 3 times and repeat a few times. Makes sure to invalidate caches if there are any outside the running applications.

Then I write down exactly how I tested it, and the slowest, fastest and average.

The next step for me is to first look for something obvious in the affected code. If there aren't anything I start adding logging to see how much time the different things take. And when I have a better understanding of how time I spent it is easy to see if there anything should be done with the code or if external factors are the cause. Like bad a database integration.