For accommodations in places most people here using sites like Booking.com for investigation, reviews, price comparisons and the map view for a better idea of where places are located in a town. I advocate then going directly to the lodging itself, when possible, to actually book but it can be a great tool.
If you're looking at staying in La Spezia for any length of time I'd recommend looking into the towns of San Terenzo (quieter) and Lerici (more resorty). They are connected by a long waterfront walkway and are probably more what you're looking for versus La Spezia.
Getting to them will require a bus ride from La Spezia since there is no train but worth the extra effort.
Hopefully you'll find what you're looking for but I think you'll need to forge your own way in some of these places,
=Tod