The Kuroshio Fish Market is across from Marina City. Although it's technically in Wakayama City, one of the main roads into Marina City is actually from Kainan, so it's closer to the sites of Kainan.
The market is said to be one of the largest fish market restaurants in Japan. One of the highlights is the demonstration of cutting large maguro. They will cut the entire fish right before your eyes and after the "performance" you can actually buy parts of the maguro that you watched getting cut. The performances only occur at 11am, 12:30pm, and 3pm.
It's a lot of fun and free to watch. Aside from the show, you can buy a lot of fresh seafood at the restaurants inside or at the market to take home.
The Onzansouen Garden is located on a peninsula in Kainan overlooking the harbor. It's relatively large; a nice strolling garden. It was first built in Osaka and then moved here by a wealthy businessman during the Taisho Period (early 20th century).
There are two small ponds and a few structures within the garden. There is also a small cave that leads you to the other side to view the harbor. The view of the peninsula from the harbor is also quite pretty! There are many lanterns throughout the garden and if you want, you can try to find the one with the hearts on it. They will tell you about it when you buy the ticket, so if you're up for it, it could be fun to try and spot it!
Entrance is 400 yen.
Kainan Station is the main station in Kainan. It is served by Japan Railways (JR).
From Osaka, go to Tennoji Station on the Loop Line and from there you can take a train directly to Kainan. Alternatively, you can take the train to Wakayama and from there take the JR Kinokuni Line to Kainan.