The walks or hikes bewteen the five villages are progressively more challenging as you go from south to north. The total hike takes 4-5 hours. But the first leg -- Riomaggiore to Manarola -- is designated the Lovers' Stroll and might be manageable for a 3-year-old. If I recall, people were using baby strollers or carriages on this leg. After that, you'll probably want to rely on the train or ferry boat.

I read recently that Cinque Terre has been designated a national park and entrance fees, used to maintain trails, are charged to hike between the towns. A one-day hiking card is reported to cost 5 euros and cover all trails, shuttle buses, and park museums, but not trains or boats. A 9 euro 'treno' card adds use of the local trains from Levanto to La Spezia, covering all Cinque Terre towns.

If you decide to reach the region by car rather than train,
Monterosso (northernmost town) probably is your best bet for your base. I suspect that if you went by train from Lucca, you would change trains in nearby Pisa and then get the local CT train in La Spezia.