I introduce tourist attractions with you
1. You should travel in Bangkok 2 day
2. Take a taxi to the port to take the boat to the island 2-3 day
3.? Take a taxi back to Bangkok to the airport and travel to Phuket 3-4 day
4.After the plane back to Bangkok. For your journey home.
but if you are tired, I suggest you stay at Bangkok 1 days. Because I'm afraid that your mom might be too tired.