"I'd sooner" and "I'd rather" are comparative terms, whereas "I wish" is a simple desire. "I'd sooner" means "before I do [this thing], I would prefer to do [that thing] "I'd rather" simply means "I would prefer".

"I wish" is for anything I might want, sometimes whimsically, traditionally in the subjunctive. "I wish I were a child again." "I wish I could go to France." It can be used in many other ways, such as to express frustration with someone without actually demanding something: "I wish you would put your shoes away" (rather than "Please put your shoes away"). It can be used in "making a wish," like in a magical context: "I wish to go to France." (Poof--I'm in France! This isn't used in an everyday context. If I know I'm not going to suddenly be in France, I would say, "I wish I could go to France.") "I'd sooner" and "I'd rather" are very similar to each other and often interchangeable--however, in the United States, "I'd sooner" is not used very often. Both are making a choice between two options, one that I would like to do more than the other. "I'd rather" is the more common phrase for choosing between two regular options: "I'd rather make cupcakes than cookies." "I'd rather go to the park than to school." "I'd sooner" could be used instead of "I'd rather," but in the U.S., would more typically be used to express that you definitely do NOT want to do something. For instance: "I don't want to live in Florida. I'd sooner live in the middle of the ocean!" (Clearly, I would not actually live in the ocean; this wild hypothetical simply expresses that I do NOT want to live in Florida.)