you can use the r software

Depends what kind of graphs you want to have. R can do most of the work, with use of different packages. Certain specific graphs have to be made by matlab like software. Decide what graphs you want to make and then narrow down the question, you will get better answers.

You can use origin 2015 or tecplot depending upon your type of graph. Both are widely used by scientific community.

for graph drawing I think that you may try graphviz. https://www.graphviz.org/

To visualize the results of scientific research, you can also use Maple.

For easy plots, even excel is good. You can carry out simple statistical analyses on it. The problem with the plots is that they do not visually appear professional. For professional looking graphs, I would suggest Origin Pro (Version 8 and above). You can carry out multiple statistical analyses on the same data too.  Another software that I would personally suggest is GraphPad Prism (Version 7 and above). The plots are very intuitive and you can edit the graphs too. The best feature of Prism is that the line and axes widths are amazing for publication.

Thank you all for your valuable comments and suggestions.