what to do in the cinque terre

Hi explorer98 -
Yours is a question that's too broad to answer without additional information. A good guidebook and local websites are also your first and best resources for browsing attractions, tours and whatnot. A couple websites to spend some time with:

Cinque Terre:
https://www.apathtolunch.com

Lucca:
https://www.discovertuscany.com/lucca/

Personally, I don't think tours are at all necessary in the Cinque Terre (I do hope you are staying there and not doing it as a day trip?) as there isn't much more to do than explore the small-to-tiny villages on foot, take a ferry ride (in season) and/or hike. Take a swim, maybe, if warm enough to do so. Folks with longer stays sometimes choose to explore a bit further afield to Porto Venere, Santa Margherita Ligure, etc. Hiking the regional park (the villages fall into this area) is one of the biggest, if not the biggest, draw for many though. Website:

http://www.parconazionale5terre.it/Eindex.php

If not hiking, your options for getting from village-to-village will be trains and/or ferries (excluding Corniglia, which the ferries don't service.)

I'm pretty sure tours aren't necessary in Lucca either but I'll defer to others on that one. Walks and bike rides along the city walls is a very popular activity.

But in the end, how to spend your time depends on your personal interests, your ages and physical abilities (the CT isn't a great place for folks with mobility challenges,) ages/abilities of any persons traveling with you, how many nights you have to work with, and time of year/tolerances for crowds (May promises to be VERY busy in the CT.) I would avoid the May 1 holiday weekend - May 1 falls on Friday this year- for SURE. Also, accommodations book up fast and early for high season so you're already somewhat behind making reservations for May.

Editing to add: regarding tours, I'd strongly advise against any sort of group tours of the C.T villages due to the heavy crowds. You stand a much better chance of escaping the peak-hour masses if you can escape to the upper reaches where most of them do not venture.

Anyway, the more you can tell us about yourselves the better we can assist. :O)
But do get a good guidebook + spend a little time with the websites for initial ideas?

The Cinque Terre is all about hiking from one village to another.
Lucca is a medium sized walled city. Ride a bike in the park on top of the city walls.Check what else interests you in a guide book.

Research by getting a guidebook, reading it and deciding if both areas would be of interest to you.

Our personal experiences in either or both may not of interest to you.

This is exactly what guidebooks are for. You will find well-researched and clearly presented answers to these (and other) questions in a good guidebook.

Definitely a good time to visit , as not too crowded and not too hot !!

I dont know of any museums in the Cinque Terre itself but there is a very nice museum in La Spezia i( Castle san Giorgio)
Also a naval museum at the edge of the port
http://www.marina.difesa.it/cosa-facciamo/per-la-cultura/musei/museotecnav/Pagine/default.aspx

We took a pesto making course inside the cinque terre which included a boat trip which was very good !
https://www.cinqueterretours.com/tours/cinque-terre-pesto-course/
The museum of statue stelle in pontremoli dating back 3000 years is in a castle and also very good , although opposite direction to Lucca!