The differences are not taught in schools but here's what I think as a Chinese speaker:
拔 and 拉 are very similar. Sometimes, 拔 implies a more greater force, which makes 轻拉 reasonable but "轻拔" sounds very unnatural. E.g. You 拔 a sword from a stone. Sometimes, instead of implying the amount of force, 拔 means the action is fast. For example, the action of pulling a SIM card out of a phone is 拔, but the action of pulling a baby's hand from the sleeve is 拉.
抽 is quite different. The action is specific. You have to have something to grab, like a handle of a drawer (the word for drawer is 抽屉), or you have to be pulling/lifting things upward. If you have to pick one thing out of many, like drawing one match out of a matchbox, it is also 抽. 抽奖 means lucky draw.
Maybe we can give better explanations if you provide examples.