I've used Sigmaplot in the past -- it is much more versatile than something like Excel. Other than that, I know R and Matlab are powerful for data visualization (among other things).

I don't know what software to suggest. Depending on the audience and purpose I would suggest you consider visiting the work of Edward Tufte which uses visuals to share data in a more meaningful way than dots, bars or lines can. http://www.edwardtufte.com/tufte/index

I'm not really sure if you're looking for plots or infographics since you've mentioned illustration. If you need something really pretty for presentation in posters, slideshow and magazines but only to plot key points you're looking for infographics software. You should take a look at links like this:

http://www.creativebloq.com/infographic/tools-2131971

On the other way around if you're looking to better the aesthetics of common plots maybe you should try taking a look at a topic like the following (research gate):

https://www.researchgate.net/post/Which_software_application_is_best_for_plotting_scientific_graphs

They talk about several plot software. The third option (best for me) is to use a good plot library in any language you may prefer. Here are some examples:

http://www.chartjs.org/

https://plot.ly/python/

http://web.stanford.edu/~mwaskom/software/seaborn/#

http://d3js.org/

http://matplotlib.org/

The best plot libraries I know of are either Python or Javascript.

The third b) option is to produce svg plots and working them directly in an illustration software such as inkscape:

https://inkscape.org

http://codeboje.de/pysvg/ (library to produce svg but I think you can do the same with matplotlib).

Hope it helps.

I can only add bokeh (a Python library) to Pedro's accurate answer:

http://bokeh.pydata.org/

Well, in fact I may add another group of tools. Visual arts languages and libraries.

Visual arts languages like Processing (a Java based language and IDE) and its JavaScript based siblings (ProcessingJs and p5.js) are complete computer languages with strong graphical libraries. They're useful whenever plot libraries aren't flexible enough (they use to have a very complete set of templates, but cannot deal with anythong out of them), but the visualization heavily depend on data making ad-hoc illustration/infographics overwhelming.

They're also useful to add interaction features to visualization, although some plot libraries have this feature too.

http://processing.org/

http://processingjs.org/

http://p5js.org/

There are also some C++ libraries like Cinder and OpenFrameworks. They are useful if performance become an issue, or if you work in a C++ environment. Some people makes prototypes using Processing as proof of concept, and then turn to OpenFramework to make the final visualization. 

http://libcinder.org/

http://openframeworks.cc/

This visualization of the evolution of Darwin's On the origin of Species by Ben Fry is a good example of works that best fit this kind of tools:

http://benfry.com/traces/

Just read an article in t3n about Lyra Visualization Design Environment (VDE). Seems to be great but is still beta.

http://idl.cs.washington.edu/projects/lyra/

http://t3n.de/news/lyra-datenvisualisierung-579247/

I have a Data Visualisation presentation in PDF format with, perhaps some useful information that you could browse through. I've attached it here. 

If you find it useful, pass it on.

QSR international's Nvivo software may be the right choice for you.

I often refer to this exhaustive list of 55 Visualization tools (can be filtered for Maps, Charts, Data & Color related tools) by Interactive things. It gives a one-line introduction to every tool, including most that Pedro & Antonio mentioned (so you can quickly glance and get an overview of the spectrum of tools that you can use).

http://selection.datavisualization.ch/

TABLEAU!!! There is a free version you can use that is remarkably easy to use and very quickly allows you to do graphing and all sorts of data visualizations from your data. You can make graphs from an excel datasheet or from online data sources. I'm sure they can handle a wide range of data resources. They have free online training tutorials as well. Plots, infographics, maps--it does it all. You can click a button and switch from a bar graph to a map of the world if you have the data points to do either. Beautiful outputs!

originpro for graphics

for illustration:

Inkscape for 2D and Blender for 3D pictures.

Blender for animations

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

One more choice is BioVinci, an intuitive, free, and easy-to-use web application that produces high quality scientific figures instantly. Just drag and drop your data. It also comprises of various statistical functions like PCA, tSNE, ANOVA tests, Linear Regression and some state of the art machine learning methods like Random forest, K-mean clustering...

You could try some examples of BioVinci here:

https://vinci.bioturing.com/panel/workset/build/k-means-clustering

https://vinci.bioturing.com/panel/workset/build/hierarchical-clustering

https://vinci.bioturing.com/panel/workset/build/tsne