Here are some tools I use in my research.  I recommend them.  Obviously, these are just my opinions; nobody’s paying me to post their stuff here.  This page is mainly for students; academics will probably know about these already.

Process is an excellent free macro plugin for SPSS and SAS, which makes it very easy to conduct mediation, moderation, and moderated mediation analyses.  I seem to use this every day.

(Also, this awesome free web tool, by Alexander Schoemann and friends, runs Monte Carlo power analyses for simple mediation models.  Invaluable!)

JASP is a very user-friendly data analysis package.  It conducts most of the common analyses you might need and, most crucially, it includes all the Bayesian versions of the analyses too.  This also gets used in all my manuscripts these days.  Also free!

Some of the JASP people broke away and created Jamovi, which looks very similar and works just as well.  It has a range of optional plugins, including the most user-friendly tool I have found for mixed models and generalized mix models.  Also free!

G*Power is the easiest-to-use free power calculator software I know of.  Unlike some power software, it doesn’t assume you have a PhD in statistics.  Free!

Surveey is an online survey service which I sometimes use for questionnaire studies.  It may or may not be free, depending on how complicated your study is.  I use it because it comes in both Turkish and English.

PEBL is a free psychology experiment programming environment.  The best thing about it is there is a large free test battery on the site, which includes fairly accurate versions of a lot of the cognitive and neuropsychological tests you might need.  Since these come with their source code, one can (fairly) easily translate the tests into your desired language, or make small modifications to the test.