– Free Software Foundation
Going through graduate studies (or even undergraduate studies), is about creativity, hard work and learning not to loose information, not to drop the ball on ideas and projects. In order word, part of it is also about being able to put your ideas to work for you in an efficient manner. In the following are a few applications to help you along the way. Of course, nothing precludes the good old pen and paper. I personally really enjoy my Lamy 2000 fountain pen and a Leuchtturm 1917 whitelines notebook (but this is for another post!).
MindMapping
For flushing out the big ideas and the relationship between them. For years, I have recommended Xmind to my students, but it is not free anymore. However, Freemind is available for macOS, Linux and Windows. It will certainly do the job.
Project planning
The Gantt Project app is available on the same three platforms as Freemind. Much better than using Excel to generate a graphical overview of your overall planning and dependencies.
Task managements
The choice for a free but fully equipped task manager, which is nice enough that topic will like to use it every day (because that it what it takes!), is not legion, unless you are willing to pay. I recommend to my students Wunderlist. It allows Projects, projects within projects recurring task, reminders, tags and so on. It is free and cross platform: macOS, iOS, Android, Kindle, Chromebook, Web, …
Another possibility is Todoist. The application has a fervent following. It is cross platform: macOS, iOS, Windows and Android (no Linux). The free version is limited to 80 projects and so advanced features from Wunderlist is available only if you subscribed to the Premium version at 39$/year.
If you are seriously considering paying for a task manager (which I think you should be considering seriously because you will use your task manager multiple times a day, every day of the year), simply buy Things 3, the price for macOS, iPhone and iPad combined is still good in the long run and out is a much nicer and efficient applications, unless you really need share/team for tasks.
Drawing-Diagram
Look at the gorgeous yED applications for macOS, Windows and Linux!
Code versionning
GitHub! If you do programing and do not have code versioning yet, please educate yourself. If you are a LaTex user, versioning might be interesting for you if you do not code. It will be interesting to see what happens now that Microsoft bought GitHub…
Scientific figure/graph generation
Here you have numerous choices foe generating professional looking figures and graphs depending on your particular allegiance. All of these will get you better looking figures than Excel. As our students regularly find out, we can spot an Excel figure miles away!
Matplotlib (Python)
GNUPlot
QtGrace
CERN ROOT (maybe an overkill for most)
And finally, GNU Octave for MatLab style analysis and data handling, though most universities have MatLab license for students. Also with Jupyter and pynotebook, there are other alternatives.
PDF library
During your studies, you will read hundreds of publications (if you do not, there is a problem either with you or your advisor, see here). This no reason for you to build a reference list manually, ever. This includes populate various metadata such as authors, journals, … 
even if you use BibTex. Most modern software will generate all of the citekey needed automatically and most will even allow you to include reference in your text as you write in application such as WORD.
Here, the best free software are clearly Zotero and Mendeley.
In conclusion
You are ready to start planning, doing, analyzing and showing your results like a pro at little to no cost for software. Like Cespenar used to say: Enjoy!
Ruminating... 