The correct spelling is "speech".
The phrase for an activity is "go (verb)ing". They are going fishing tomorrow, or, they are going to go fishing tomorrow. The difference depends on how concrete the plans are.
Now we have a small problem. Is the "tomorrow" in the direct and reported sentences the same day? Think about how much later the speech is reported.
"They said that they were going fishing the next day." This is technically correct, especially if you are reporting what they said several days later or more. If you're still struggling with this form, don't confuse yourself with unclear examples.