Hello Veronica. I draw illustrations for the mycological lab and UFRN herbarium in the botanical department of Federal University of Rio Grande do Norte, Brazil. Is not a simple way to become a illustrator, need some training and practice, but is not impossible. If your question is about how one person can learn, the simple answer is "try". Is necessary some practice to define the heights and depths in a image. Three techniques can be used, two based in camara lucida style - for begginers and advanced researchers - , and one in observation - more complex, recommended only after some training. The "camara lucida style" can be used (1) in the equipment with the same name attached to a stereomicroscope or optical microscope, or (2) using a photo printed in the size you want, and after this, using a parchment paper to cover with nankin. These two techniques is good to mantain the proportions of the sample you want to draw. The third technique is more complex, and sometimes will be a reflex of the illustrator style, but you don't need any specific equipment, only a paper, pencil/pen/nankin and patience. It's possible to combine the camara lucida with the observation (sometimes is more easily to do): you can take the dimensions with the camara lucida or a photo, and complete the draw based in observation, adding some shadowns and other structures. All of this is for hand-made draw illustration.

To draw in a software you can use the same techniques, just scanning the structures to take the dimensions, or drawing directly. Is recommended use a graphic tablet to this part, once using the "mouse" is a hard way to conclude. Softwares like corel draw, photoshop and illustrator are recommended to draw black/white illustrations (but I recommend the use of manga studio to take more control in the graphic tablet's pen and the possibility to rotate the page more easily than the others softwares). Sadly these softwares aren't free. GIMP (to draw) and ChemSketch (good one for graphics and chemical structures) are free options, but if one want to enter in the professional way is better take the other non-free software (with the exception of the ChemSketch, this is really a friendly program to design some graphics). For color illustration you can use photoshop, illustrator, manga studio or GIMP.

About the third question, yes. I started working in my own papers, but after this other people search me to start services, like new plant species, revision of fungal herbarium, book plates, and so on.

Thank you so much Rhudson, very detailed answer. This was very helpful. 

you can also try:

Inkscape for 2D and Blender for 3D pictures.

Blender for animations

For instance: https://www.youtube.com/watch?v=RasemoYzsFA