You'll need to use "going" for most of these uses. For example: "I'm going to shop at that store." "Are you going to visit grandma?"

If you say "I will go swimming today", or "I am going swimming today", then it is just a colloquial way of forming the future tense. If you say "I go to swim", then it is a strange way of saying you are about to make a journey in order to swim. However, you might say "The policeman went to make an arrest." It implies that the policeman made the journey with the determination to carry out his duty. To say "I go do to X" implies a strong sense of purpose and that X is important. It is therefore likely to be pompous, but one can imagine circumstances in which a top diplomat or government official might say it.