First, start your website and portfolio now. This is the main thing you’ll be showing potential clients, so it’s good to get started early. It doesn’t need to be fancy, but good images and your own domain name go a long way.
Second, join the Association of Illustrators. If you come from a science background, you might not know a lot about the business of illustration. The Association of Illustrators, or AOI, is a very useful source of advice on contracts, pricing, and copyright.
Finally, don’t be afraid to try out new things. There’s no set path to being a science illustrator, and it’s good to have your own projects to see you through the natural ups and downs of businesses *.
* When I’m not illustrating for researchers I’m designing science posters to sell online, tutoring chemistry, and creating new patterns of neurons to print onto dresses.