Mamabrio,
As I recall, the boat doesn't stop often in La Spezia so the timing may not work. Your best bet is to take the train from La Spezia to Riomaggiore (about eight minutes) and go from there. Note that the boats are seasonal and operate only in good weather. If they're not running, just use the local trains to travel between the five towns (they're cheap).
Regarding "walking", I'm not sure the trails will be a realistic option for a group of 12. Only two of the Sentiero Azzurro trails are open at the moment (although I haven't checked recently) and they're the most challenging two of the bunch. Your walking will probably be limited to just exploring the towns on foot.
I'd suggest leaving Monterosso for last and having supper there, as there are a lot of nice restaurants, and you'll need a larger place to accommodate a group of 12. You might try Cantina di Miky (new town, close to the station - you may have to sit inside with a larger group, depending on how crowded the outside patio is), L' Alta Marea (old town, a short distance up via Roma - same comments about sitting inside) or Ristorante Belvedere (old town, through tunnel and right on the beach - get there early as they tend to fill up quickly).