'I'm joking' and 'I'm kidding' both mean 'I'm not being serious'. They have the same meaning. The only difference is that 'joking' is standard English, while 'kidding' is colloquial/informal.
'Teasing', which is often transitive, is always directed at a specific person. When we tease a person, we are trying to get a reaction out of them by playing upon that person's particular weakness. For example, if you know that your friend is scared of spiders, you might tease them. So, if you say "I've got a spider in my bag" ( when you haven't), you are joking/kidding about having a spider in order to tease your friend.
Teasing does not have to be the same as joking, and it can be either kind or cruel. A good friend of mine here on italki knows that I'm not very good at maths, so he often makes reference to mathematics in his posts to me, as a way of teasing me. This is friendly teasing. But if children regularly make jokes about a fat classmate's weight, this is cruel teasing.