This is the pudding recipe I have used for years. It works with any "milk". I always used cow's milk, but now that I am forced to eat vegan I use whatever milk I am drinking (currently hemp milk). This is for chocolate pudding, but you can leave out the cocoa and put in whatever flavoring you want, eg more vanilla, lemon juice. Not sure what you would do for banana pudding, maybe you could google non vegan recipes for banana pudding and see how they get the banana flavoring, that is how I figured out how much lemon juice to use for lemon pudding.
1/3 c. sugar (vary depending on taste, original recipe called for 3/4 c)
1/3 c. cocoa (vary depending on taste, I actually make this heaping)
1/4 c. corn starch (we have corn allergies, so use tapioca starch as sub)
pinch of salt
2 c. "milk"
1 t vanilla (this is optional for the chocolate pudding)
Mix dry ingredients in sauce pan. Pour in "milk" and heat on medium heat stirring often. Once it starts to thicken, lower the heat some and continue to stir. Once it gets to the consistency you want (actually good for it to be just a hair runnier that you want it to be), take off the heat and stir in vanilla. Cool in refrigerator uncovered...Enjoy
For vanilla pudding put in 1 Tbl of vanilla.