Marie, you don't need any sort of tickets; you need a park pass to hike the segments of the Sentiero Azzurro (Blue Trail/SVA2). Unfortunately 2 of the 4 segments have been closed for some time (landslide and rockfalls) and will remain so through 2017.
Those stretches are between Corniglia - Manarola (592-2) and Manarola - Riomaggiore (592-1) The sections between Monterosso - Vernazza (592-4) and Vernazza - Corniglia (592-3) should be open but heavy rain or sudden change in stability can close any of the trails at any time.
These are not the only trails in the park - there are many others - but they're the most traveled, when open, as others tend to be longer/more strenuous. You can view a map of the trails and which are currently open/closed on the park website:
http://www.parconazionale5terre.it/Eindex.php
http://www.parconazionale5terre.it/Esentieri-outdoor.php
The red-and-white circles and black lines indicate trails which are closed. Clicking on the individual listings below the map will provide a description of each trail. You can also get descriptions by clicking segments on the map itself.
And here is info on the pass for the Sentiero Azzurro. You can buy it at the train stations in any of the villages, at kiosks at the entrances to the trail, and you do not need the pass for trails other than the two segments of the S.A. listed above.
http://www.parconazionale5terre.it/Ecinque-terre-card.php
Getting around by train is easy and you might want to take at least one ferry ride. If taking the trains about, do NOT forget to validate your tickets before boarding as fines are high if caught with unvalidated tickets, and the authorities tend to be especially vigilant in this area.
Monterosso is our favorite but to be honest, at this point I think you'll have to take anything that's still available in ANY of the villages as accommodations fill far in advance of high season. If you can't find anything, try Levanto or La Spezia; only a few minutes away by train. The CT is going to be very, very busy in July...