Travelling from Kyoto to Osaka
Kyoto is only about an hour away from Osaka by train, and even faster if you take the shinkansen. It's fairly cheap too, by regular train it's generally about 500 yen if you take the Hankyu line and a little more if you take a different company's. Would not recommend taking the shinkansen unless you have the JR pass because it's significantly more expensive and not that much faster (15 minutes instead of an hour?). However, if you have a lot of luggage, note that the regular trains usually do not have dedicated luggage storage space unlike the shinkansen, so take that into account.
Quick Introduction to Osaka
I spent about two years living in Osaka and it's my favourite place in Japan. Some people prefer Tokyo, some prefer Osaka, but you won't know which type of person you are unless you've been to both.
Osaka generally is divided into two areas. The area to the north is the Umeda area, and that's where you'll find all the highrises and department stores. It's a lot more modern than the area to the south which is the Namba area. The Namba area is a lot older, run-down, but has a lot of great character. You should absolutely visit both, but if you had to visit only one, I would say go to the Namba area since Umeda isn't too different from most modern Japanese city centres and you can get a similar experience in Tokyo.
Umeda: There's a lot of good restaurants in Umeda, and a lot of places to shop for clothing. Umeda is where Japanese people go on weekends to have a meal with their friends, go shopping, or have fun at bowling or karaoke. Umeda is generally a little more expensive than Namba in terms of everything from eating to accommodations. The only foods that I would recommend eating in Umeda over Namba would be ones that have chain locations like conveyor-belt sushi (my favourite chain is Kurasushi) or ramen (Ichiran, for example). There's a large Don Quijote in Umeda (see Namba section for more details) and a Yodobashi Camera for electronics and games.
Namba: The famous Dotonbori area and Shinbaibashi shopping street is located in Namba. Lots of great food. Definitely try out takoyaki, okonomiyaki, ramen, and kushikatsu. A note for those who are interested in trying out Ichiran ramen, since it's a pretty popular restaurant for tourists to try-- there are multiple locations in Dotonbori alone and they all taste the same. Some people will line up at the big store on the main canal street but it's not worth that extra lineup. Go to one of the side locations and it'll be just as great. Same thing for takoyaki, it's not worth waiting in line for 30 minutes for one store if the one next door has no lineup. Some of the more popular stores for shopping are Don Quijote for souvenirs and snacks (chains all across Japan and in Umeda too, but the main store in Namba is pretty big), and Bic Camera for electronics or video games. Bic Camera is basically the same as Yodobashi Camera.
For anime lovers: Nipponbashi/Den-den town is also in Namba. It's not as impressive as Akihabara, but there's still a large Animate (anime-goods store) and a variety of other stores.
Shin-sekai: Even more to the south than Namba area is the Tsutentaku area of Shin-sekai. Generally known for the kushikatsu, but honestly you could get that in Dotonbori too (they're usually chain restaurants). Tsutentaku was pretty cool to visit though. Shin-sekai is even more run down than Namba, unfortunately shin-sekai has a bad reputation of being the "slums" of Japan. There is some truth in this. I have travelled all across Japan and this area is one of the most sketchy areas I have ever been to... and lived in (I spent about 6 months living near here). That said, it is not unsafe, especially during the day. There are some homeless folks but they keep to themselves, and a large population of labourers. While living in this area, I generally avoided going out by myself at night, but if you're in a group it should be generally fine. The worst that would happen is probably someone yelling incoherent drunk ramblings at you.
Back to this itinerary
On this particular trip, we checked into our AirBnB in Osaka and went shopping and eating in Dotonbori and Shinsekai on Day 4.
Whenever I'm travelling (and not living) in Osaka, I rent an AirBnB near Dotonbori. Generally, you can find a really reasonable place within 10 minutes walk of Dotonbori or Shinsaibashi. I have never booked a hotel in this area, just because the value for the price isn't worth it.
Made a day-trip to Nara. See the Nara part of the trip.
We visited Umeda for this day.