I've been playing since the end of season 2, beginning of season 3, and I still play bots. I don't do it to get better at beating bots, but to work on the feeling of certain champions or builds. I just like the care free space in bot matches, but sometimes there's a few people who are big jerks and say, "OMG KS, uninstall."
Back to your question about how much time-wise it takes to graduate (I'm going to refer to time as levels.)
levels 1-10: Basically Only bots, sometimes did PvP, but this was the time when duo top was much more popular in the lower levels.
levels 11-20: I've made some friends, and I'm getting good at destroying bots, and now I want to test my main champions on real people.
levels 21-29: I start to study masteries and runes and gameplay strategies online, MobaFire was my favorite website for learning the game.
Level 30: I hit level 30 within season 3, and to be honest I wasn't really excited about ranked games, I mean normal games was enough stress for me already. To be honest I play normal games as much as I play ARAM, and I play bot games also somewhat evenly too.
Conclusion: I didn't mean for this to be such a long reply, but I guess I really got into the subject. I'd say you can learn some stuff still from bot matches, but you learn a lot more from PvP, and even more in Ranked games. I Usually Only try to play normal games with friends to have more fun, cause that's why I play this game.