Osaka Castle.

No so obvious things:

Aquarium

One of the largest in the world. Official name is "Kaiyukan", which means "Playing in the Sea Pavilion". Plenty of others play too, so the best time to visit is mid-afternoon during the week, to avoid the school groups and weekend crowds. Usually open 10:00-20:00.
Address: 1-1-10 Kaigan-dori, Minato-ku, Osaka City 552-0022
By subway: take Chuo Line and get off at Osakako (Osaka Port) station. 5 mins walk to Tempozan Harbor Village, which is centred round the Kaiyukan.
Entrance: Adult (16+) ¥ 2,000

Suntory Art Museum

Designed by Tadao Ando, this building itself is as interesting and impressive as the exhibits and Imax screen inside. It's in the Tempozan Harbor Village, close to the Aquarium (see above)
Address: 1-5-10 Kaigan-dori,Minato-ku,Osaka 552-0022
Entrance: Adult ¥1,000 for exhibition, ¥1,000 for Imax, ¥ 1,600 for pass including both.

Dotombori

(or Dōtonbori) the main area for bustling night-life. Dotombori-gawa (the canal) and the pedestrian arcade alongside it are brightly-lit and lively in the evening. Dotombori bridge (the footbridge over the canal) is a great place to watch Osaka go by.
Address: between Dōtonbori-bashi (bridge) and Nihonbashi Bridge in the Namba district.

America Mura Street

An odd little area of young Japanese putting on American style. Interesting shops, outfits, and bargain vinyl records.
Closest subway station: Shinsaibashi, Midosuji Line (red).

"Moss Buddha"

A charming Buddha statue and votive pool tucked away in an Osaka locality. Not easy to find: follow Shinsai-bashi-suji south of Dotombori Bridge, up to the Glass Factory store, then turn right into a cobblestone courtyard. Locals pray and splash water on the Buddha from the pool in front. Out of respect, the locals prefer no photography.