Osaka has plenty of great districts for browsing restaurants and eating.
In the north (Umeda area), there's Kitashinchi, with a wide variety at a range of price points (many of the top-rate, high-end places are here).
In the south (Namba area), there's the famous Dotonbori area, and the Hozenji-yokocho area. Also, be sure not to overlook Shinsekai and the Koreatown at Tsuruhashi!
Here's what springs to mind:
1. Takoyaki (Octopus balls)
* Ajiho (my favourite)
2-8-33 Nishi-Shinsaibashi, Chuo-ku
In Amerika-mura, walk from Shinsaibashi station
! Also has a great doteyaki (simmered tendons) and oden (simmered fishcake) stall.
! If you order takoyaki inside, they'll give it to you with a dashi dipping sauce like akashiyaki.
* Wanaka (popular)
11-19 Namba-Sennichimae, Chuo-ku
Next to Douguyasuji, walk from Namba station (although there are branches elsewhere).
2. Udon (Wheat noodles):
* Kamatake Udon
4-20 Namba-Sennichimae, Chuo-ku
Not far from Douguyasuji, walk from Namba station (although there are branches elsewhere).
3. Horumonyaki (Grilled offal)
1-10 Shimoajihara-cho, Tennoji-ku
In the Koreatown, walk from Tsuruhashi station (although there are branches elsewhere).
! Recommended is the "urute": pig throat.
4. Kushikatsu (Crumbed skewers):
* Daruma (the most famous kushikatsu place, if not the best)
2 3-9 Ebisu-higashi, Naniwa-ku
Under Tsutenkaku tower, walk from Ebisucho station (although there are branches elsewhere).
* Chibo (the standard travel-guide pick)
11-27 Namba-Sennichimae, Chuo-ku
Walk from Namba station (although there are branches elsewhere).
6. Yoshoku (Japanese-Western food)
2-2-1 Yariyamachi , Chuo-ku
Walk from Tanimachi 4-chome station.
! Haven't been here, but a friend recommends it. This restaurant is the ancestor of the 3-Michelin Star "Fujiya 1935."
7. Kappo (Counter-dining)
* Geppa (my favourite)
1-4-25 Dojima, Kita-ku
Walk from Kitashinchi station.
! The "picked vegetables" lunch set ("Natsumi-zen") is Y1000. Their dinner is rather more expensive, but it's one of my favourite restaurants anywhere. You can see the lunch here:
... Osaka is also famous for fugu (pufferfish) and for kappo (counter restaurants), but they don't really fit the bill of "casual" nor "budget." I don't recommend eating budget fugu at Zuboraya, it's pretty bland there.
What sort of dining are you looking for? High end, casual, etc?
Osaka is famous for okonomiyaki, and I've had very good experiences at Kiji for this dish. Its not entirely easy to find, but its under the Umeda Sky building which is pretty easy to find given its massive.
But there are many places to get okonomiyaki, yakisoba is also popular in places like that.
How long are you there? How much money are you looking to spend? etc? Where is your hotel or where are you staying?