Hi Virginia -
The unfortunate news is that the CT - pretty much ALL of it - will be busy during the daytime in June. Corniglia is the less visited as you must take a little bus, climb a steep road or a loooooong flight of steps from its train station up to the village - obviously options # 2 and #3 are not possible in your case - but because it's so tiny it also doesn't take many bodies to overwhelm it. We were there early in the morning before it got busy but people I know who've who visited middle of the day/early afternoon said they found it uncomfortably crowded. But maybe others have had different experiences?
And just as unfortunately, the Via dell' Amore path between Manarola and Riomaggiore has been closed for some time and will remain so throughout this season.
Is this a day trip from Lucca or are you planning on staying overnight? If planning on staying, I'd recommend Monterosso as it has the largest area of relatively flat ground and is the largest of the 5 villages so is better able to disperse the daytime crowds. Corniglia would also be OK - quiet in the mornings and evenings - but I'd be very careful of booking something easy to get to from the little bus stop; a lot of accommodations there (in all five villages, in fact) can involve a lot of steps to get to. Having to navigate fairly steep inclines is also common to parts of almost all of them.
There are many trains which will take you to the CT so it's best of explore them on the Trenitalia website:
http://www.trenitalia.com/tcom-en
Just choosing Lucca as a departure station and Corniglia for arrival, it looks to take anywhere from 2-3 hours one-way from Lucca, with train changes in at least 2 different places depending on the train you want to take. If deciding to start in a different village, just use that one for your arrival station. If going back to Lucca that same night, just change the names of your arrival/departure stations. You do not have to purchase tickets in advance; buy them in Lucca on the day of your journey, and validate any which are on regionale trains (no seat assignments) before boarding. You'll need to validate train tickets for any trains you take between villages in the CT as well.