Some of my friends who live in Japan do take trains long distance, they tend to be comfier, but my friends who use to go from Tokyo to Osaka a lot to visit one another and lived in Japan tended to fly. If you want to be as cheap as possible you can get low fare flights for like 4000 to 5000 each way on Peach or one of the other discount airlines and do KIX to NRT. You can take a bus from downtown Osaka to KIX for like 1000 yen and there are 1000 yen buses from NRT to Tokyo. There are discounted fares on JAL and ANA as well.
Since I am old and can't sleep on the bus, I prefer not to do overnight buses. I do have friends who were studying at Kansai Gaidai, who used the overnight bus. I think it was hard to get in all the touring we had planned for when they arrived. But that really depends on you, I'm old and now travel with a kid, so I like fast and comfortable.
Itami is a great airport. I flew into it from NRT once. I did NYC to NRT to ITM. The transfer was easy and I had way too much time. I should have gotten more snack before going through security. Haneda, Narita, and KIX get a lot of flights from overseas, or at least did pre Covid so they all have a lot of English. If you've flown before, flying in Japan is not all that different. There is a lot of English signage at least at all the airports I've used in Japan.
Me, I would prob just take the train, I might do the Tokai offer or just buy in advance or see which was cheaper. A JRP is great if you are doing enough travel or just want the freedom to go where ever but one disadvantage is they do not cover Nozomi, just Hikari and Kodama. There are on average 6 Nozomi an hour from Tokyo to Shin-Osaka, 2 Hikari, and 1 Kodama. Nozomi are not only the fastest trains, but they are the most frequent. If you have a lot of other travel planned, then they can be worth it, but since you're only planning week ends it doesn't seem worth it.
Assuming you are arriving into KIX, this is a good deal and gives you a refillable IC card.: https://www.westjr.co.jp/global/en/ticket/icoca-haruka/
The Welcome IC card currently being offered by JR East is not a good deal for you since it is only valid for 28 days. I got one my last trip because I had lost my old IC cards and then just ran it down buying snacks at Narita before I left.
There are lots of smaller passes you can get, some in Japan for not that much money, that could be worth it depending on your exact plans, since paying a little more, but having the flexibility to not use a pass might save you in the long term. I love most of the JR West passes since they include the Nozomi. I've also used the various JR East passes which are a great deal but since Tokyo is not your departure point, they're not so useful.
If you're coming from the USA, I would look into what your phone plan here covers in terms of roaming. I am unsure how easy it is to get a phone for just a few months outside offers aimed at tourists and those tend to be more expensive. I know in my case, my roaming charges are a lot less than renting pocket WiFi or a Japanese SIM card, even on shorter trips like my last one. I do find sometimes my texts to Japanese numbers do not arrive. (I could be doing something wrong) I tend to use Line to contact friends in Japan and have yet to have any issues. I'm on Google Fi and use a Pixel.
I've not taken a bike from Osaka to Kyoto but there are lots of cheap and plentiful trains Whether you use JR, Keihan, or Hankyuu will depend on your starting and ending point.