Like the others have said, there's no fixed time on Bot games before normals.
What I like to do, and I can recommend (and I'm rather new to the game, 2 months and lvl 20) is:
Pick the champion you like, hit mobafire.com and search for builds for that champion. Take a look at a few (make sure they're fresh for current season) and choose the one that fits your playstyle (Read the guide, at least a quick read, don't just copy items).
Do 2 or 3 Bot games, see if you are getting it right. For instance, if you're playing Statikk Shiv Yasuo you should have it by no later than lvl 10. If not, you're missing too much CS. Same goes for all champs. Check out your main item and look at some YouTube videos. At what lvl are the "pros" (actual pro or high elo) getting that item?, and you?.
At least for me that's a good indication of how good or bad I'm doing on CS.
And the last advice I can give is...download LoL Replay and use it. when you go to a normal game and you get facerolled...go to the replay. What happened?, why you didn't see that gank, why you couldn't do enough damage to finish off that champ, etc.
Same when you snowball hard. What you did right and make sure to remember it.
I only have 40sh victories, I'm no pro. But this has helped me get a lot better with the champs I use (Yasuo is on that list, but I'm still not good enough with him to carry a game, and most of the time I need to be :P)
Hope this helps.