Take the Nankai Rapi:t alpha or beta train is the fastest way to get to Osaka from the Airport. Once you exit the departure hall, the train station is literally just a bridge across. The price is just 1300 yen or something. Btw, the kaisoku train isn't that fast, it stops at every station and you really will waste a lot of time if you do that.

Osaka station is operated by JR, other companies stop at Umeda, which is pretty much in the same area but requires some walking (8mins or so) but nothing that's hard to handle. If you are taking the train to Kyoto, make sure it's SHIN KAISOKU (新快速）. There are 2 lanes on the same platform. One of which indicates KAISOKU and the other for SHIN KAISOKU. Just queue behind the other people for SHIN KAISOKU and you'll be fine.

Depending on how you wish to travel, getting a pass isn't always the cheapest thing to do. You need to do your homework thoroughly on how much you will spend on transport and learn the subway lines. A typical subway ride from honmachi to namba cost about 210 to 230 yen. From Namba to a place like Sakasuji Honmachi probably cost about the same but may require you to change trains, but as long as you pay the correct fare, there's no problem jumping from a train operated by JR to another operator like Hanshin.

While in Kyoto, its better to walk. Learn and google up walking guides. Only places like Kinkakuji and Arashimaya Tenryuji will you actually need to take a bus or JR as they are actually pretty far from Kyoto station. I don't recommend walking there either lol.