thomas,
No, it's not a good idea to stay in Genova as that will add considerably to your travel times and costs while exploring the Cinque Terre (I assume that's what you're referring to as the "Italian Riviera"?). Staying in one of the five C.T. towns is a much better idea, and will provide a much more pleasant and enjoyable travel experience. There are lots of accommodations listed in the Italy guidebook, or you can check Trip Advisor or other sources. I prefer to stay in Monterosso as it's the largest of the five towns and therefore has the greatest selection of lodgings, restaurants and other amenities, as well as the best beaches and a full service rail station. If you can't find anything you like in the five towns, staying in Levanto is also a good option as it's only about four minutes by train from Monterosso.
At that time of year, the Cinque Terre will be BUSY so I would highly recommend getting a hotel booked as soon as possible. I usually start booking accommodations in February for travel in September, and even starting that early it's sometimes difficult to find a vacancy at the hotel that I prefer.
Booking train tickets in advance probably won't be possible, as many of the trains that operate in that area are Regionale. It's very easy to buy tickets when you arrive there, either from a staffed ticket office or a ticket Kiosk. There are a few potentially expensive "caveats" to be aware of when riding trains in Italy, and if you need more information post another note.
"Is it walkable from the train stations along the Riviera to the town centers?"
All the towns are very walkable and the stations are right in the town centres (except for Corniglia, which is located on top of a rocky bluff). You may find it helpful to watch THIS short video to get some idea of the layout of the towns.