Aeolian Islands

The most straightforward way it to take a hydrofoil from Milazzo:

https://www.thethinkingtraveller.com/thinksicily/travel-to-and-around-sicily/travel-to-the-sicilian-islands/aeolian-islands.aspx
Milazzo is the main departure point for the Aeolian Islands with several hydrofoils leaving daily throughout the year. Crossings take between 45 mins to 3 hours depending on which island your are heading for.
Liberty Lines, tel. +39 0923 873813
Siremar, +39 081 4972999

Liberty and Siremar ferry links with schedules are found in the above link.

I cannot be of much help but believe for visiting as a day trip your best options would be from either Milazzo or Tropea.
There are boats from each of these spots to the islands.

The 2 towns mentioned are not close to each other, so I imagine it is not a choice between them just which ever is closer depending on your other plans.

Milazzo is a little closer to most of the islands in the chain except to Stromboli which would be closer to Tropea.

Thanks forum friends.