"Hike" to Portofino (it's very easy, more like a stroll). At Portofino you can hike above and around the town - not hard. If you are up for more serious hikes, there are some moderate to difficult hikes you could take from there around the peninsula. You can hike to the monastery at San Fruttuoso or up to the town of San Rocco. (There are two alternate trails; one of them, via Batterie, is more scenic but difficult, in spots requiring you to hold on to chains to scale across part of the "trail" with no footing. That's the one I hiked.) From San Rocco it's basically a few hundred steps down, down, down to Camogli, a really lovely town to explore, probably pretty busy with Italian tourists in July. There is a train between Camogli and SML. There is also ferry service between Rapallo, SML, Portofino, and San Fruttuoso. (Maybe to Camogli too? I don't remember.)
There is great train service up and down the riviera between Genoa and La Spezia, so there are lots of little towns to check out, including big Genoa if you want. The Cinque Terre will surely be overrun with tourists in July, but a lot of people would want to day trip there.