Day 1 — Castle, then Namba after dark
Start at Osaka Castle in the morning, while the park is coolest and quietest, and take your time in the Nishinomaru Garden if it's cherry blossom or autumn foliage season. In the afternoon, head south to Namba and Dotonbori — walk the canal in daylight to see the shopping arcades properly, then let the neon take over as evening comes in and eat your way through takoyaki and okonomiyaki along the way.
Day 2 — Shinsekai, Tennoji, and the bay
Spend the morning in Shinsekai and Tennoji: Tsutenkaku Tower, a kushikatsu lunch in one of the alleys, then Shitennoji Temple or Abeno Harukas's observation deck depending on whether history or a skyline view appeals more. In the afternoon, head to Osaka Bay for Kaiyukan Aquarium and the Tempozan waterfront, or swap this for a full day at Universal Studios Japan if that's a priority — it doesn't comfortably combine with anything else on the same day.
Day 3 — A day trip out
Osaka's location makes a day trip the natural way to spend a third day — Kyoto and Nara are both under an hour away by train and cover very different ground: Kyoto for temple density and traditional streets, Nara for the free-roaming deer park and the Great Buddha at Todai-ji. Pick one rather than trying to combine both in a single day.
This itinerary assumes a central base in Namba or Umeda, both on the Midosuji Line — it keeps every day's transit time short enough to fit everything in without an early start.