If I would like to see if the random assignment works well, which t-test should I use?

good ^^

Ok.

T-test is powerful in testing the group difference, but I don’t think the T-test can help you determine whether the random assignment is truly random. I suggest the following three ways to help you assess the randomness of the assignment. 1) Visually compare the distribution (box, histogram, etc.) of each variable in group A and group B. The more similar they are in appearance, the more likely the assignment was random. 2) Statistically compare the distributions of each variable in groups A and B using a goodness of fit test. Two common ones are Kolmogorov-Smirnov Test for continuous features and Chi-Squared Test for categorical features. 3) Build a simple model to predict which group an observation will be assigned to. If an assignment is truly random then you should not be able to easily predict which group the observation was assigned to.