These are all of the different ways I can think of (I might have missed some) of how to request something in Japanese in no real particular order:
After some time of learning these conjugations I stumbled across various sentences, usually in video games or recipes, that would use the base dictionary form of a verb, rather than the above, to ask me to do something. I have the feeling it also applies to telling myself to do something (in an alarm for example) because saying 牛乳を買って feels a bit odd as opposed to saying 牛乳を買う, but I'm not entirely sure.
Here are a few more examples:
I imagine the latter doesn't really have much to do with requesting at all and is more just used for instructions, but I'd just like to have a more concrete definition of the difference/usages.