Yes, you really have to do all those tests.
Much of statistical analysis is based on the assumption that you have a lot of data to work with and the data follow a normal distribution. Beyond that, though, there are reasons we can’t just work up a t-test and a Pearson correlation without examining the data set first.
Non-parametric statistical tests are appropriate in different instances. This article from Boston U School of Public Health is a good place to start if you’re wondering.
In any instance, it’s always a good idea to run a normality test (Kolmogorov-Smirnov or Shapiro-Wilk, depending on sample size) before further analysis.