Kobe Station to Osaka

Quick Q: What is the best and convenient way to travel from Kobe Airport to city in Osaka?

Quick A: Hyperdia is your friend.

Taxi, if you have enough money.
https://www.taxifarefinder.com/jp/

If want to know railway, Google enough.
https://goo.gl/maps/GYA9E6pqcs42

Convenient and best are not mutually inclusive.... Convenience may take more time and best may come at a price. Better question is: What is Osaka to you? The station or the city?

Third option down in a google search

https://www.kairport.co.jp/eng/access/pub_train.html

I haven't done the route. but Portliner and JR Kobe-Kyoto Line seem inexpensive and quick enough for my liking.

from airport to sannomiya station you can use the portliner, after you can take JR (410Yen) or Hankyu (320Yen) they are both cheap, but the JR is more fast than Hankyu.

I recommend another route as below:
Kobe Airport-(Port Liner)-Sannomiya-(walking)-Kobe Sannomiya-(Hanshin Railway)-Osaka Namba
The fare is 740 yen in total and it takes 1 hour and 10 minutes at fastest.