Never been to Rapallo, but have been to Camogli, Levanto, the Cinque Terre, and Genoa.
We've been to Genoa twice, and enjoyed the city very much both times - there's something there for everyone. As Ian mentions, a very nice covered market, Mercato Orientale :
http://www.mercatidigenova.it/mercati/orientale/
and of course all types of food shops and businesses in the city, from seafood and pasta to gelato and chocolate (look for Viganotti Chocolate and Gelato) . There are also small weekly markets, such as a biological one, that are pleasant too.
The old city and its maze of alleys is a great area to explore, especially in the evenings, and manages to surprise with every turn. Beautiful architecture, from palaces to cathedrals, and all these with the backdrop of the port. New and old don't clash - the place has character.
The main street Via XX Settembre, and the area surrounding the market, are attractive for window shopping.
The promenade with the windmills in the port is also a nice place to soak up some sun and have a leisurely walk, if the weather's good. There's a lovely observation point on the city - the Spianata di Castelletto - that can be reached by a funicular, or by a meandering stroll.
We liked Camogli and Levanto also, and are definitely worth a visit. Each have their special charm, Camogli being postcard-perfect, while Levanto is a town you'd like to live in.
The Cinque Terre we visited in the beginning of November, so we had the place almost to ourselves - I can't imagine being there during times when the villages are more populated with tourists. They are definitely also worth a visit - but as strikingly beautiful as they are, we never felt a need to re-visit.
Another area that we have explored and enjoyed, and would definitely return to, is the mountainous area north-east of Rapallo : the valleys of the Vara and Taro and surrounding mountains. Off the beaten track for most tourists, but with no lack of nice villages to stop in and examine, plus the beauty of the mountains. A car of course is a must. Varese Ligure, Borgo Val di Taro, and Bedonia to name a few of the towns. If you can time a visit there with a local festival, then even better.