You should drive the transition since you are going to be the one responsible for the work in the future. As far as whether to let the dust settle before starting the transistion, you should talk to Bob's old boss about this, he will have a better feel for his personality and know if he is likely to be more cooperative after he has a chance to calm down or if he will get more upset the more he thinks about it. This person can probably give you some insight into how to approach Bob. He may even want to do the set up of the intial meeting.
You shouldn't make the assumption he is going to be unhappy about this. He may be glad to escape into a new role. It is usually better for your relationships to assume something positive about someone rather than something negative. Your attitude towards that person will reflect how you view him and he will know if you are thinking, "Hey good for Bob he is moving on to this exciting new task" or 'Ugh, Bob is the only who knows this stuff and he is getting canned for being so bad at it, this is going to be a horrible meeting." Additionally even if he is unhappy, most people will recognize that it isn't your fault that he is being moved and that if he wants to stay at the company or leave on good terms, then it is in his own best interests to help you make the transition. So I would, intially at least, assume he will try to cooperate.
Now you will be free to do things your own way once you make the transistion. Your job right now is to find out how things are being done right now; it is NOT to criticize how they are being done right now. Think about it as if you were the one in Bob's position. How much more information would you give to the person who is respectful of your expertise and is listening and taking notes than to the person who is saying, "But in modern practice, you should have been doing..." every other sentence. He did do this job for a long time, he is the expert on what was done even if circumstances are changing. Try to get some history for why things are done the way they are. You might find that there are roadblocks such as government regulations that you were unaware of. Be careful how you phrase this type of question though. There is a world of difference between, "Why on earth did you do that?" and "Can you explain the history of this process, how did it evolve and what sort of limitations did you have at the time that dictated the process design?"
It is not unreasonable to ask for a written transistion document. But don't rely just on having such a thing. Talk to him directly, take notes (even if you know you are getting or have a transistion document) and then go back and think about what he said and see if you can think of any missing information. He may not be witholding deliberately, but when someone has done a job for long time, they may do some things so automatically, they never think that someone new might not know about them. So actively look for things you want to ask about what he told you. If possible, observe him doing the job and not just have meetings.
If he is fully cooperative, make sure that his old boss and his new boss know that. Knowing that you will give him credit will help you get that cooperation two months from now when there is an urgent thing that you need to ask him how to handle because neither of you thought about it during the transition. (Make sure to ask what types of urgent things tend to come up and what he does about them. Sometimes people tranisition only the ordinary stuff and don't think to tell you about how to handle it when things go wrong.)
And don't publicly talk about how he messed up and you got the job. This is not the way to get the future cooperation that you may very much need when you actually try to do the job.
Now it is true there are some bitter people who won't cooperate. In this case, you need to get the next level of management involved fast. If, for instance, he turns down your meeting requests and won't suggest a better time, then don't wait weeks to escalate this up the chain. Have his manager make sure he schedules the meeting. In this case, you might even want the manager to attend the first session. If he isn't coperating, you need to have it documented that you tried to get his help or you may well receive the blam in three months when something goes wrong that only Bob can fix.
Now besides getting a transition from him, you need to talk to the people who wanted him replaced and get a good idea from them where you need to make changes. What you think might have caused him to be moved out of this role may not be the whole story or even the most important thing.
If he had subordinates, you need to talk to them too as part of the transition. If he is not forthcoming with information, they can probably point you to much of it. Further, they are probably nervous about this new person and how is he going to change my job and they may genuinely like and respect Bob and be angry about the change. You need to deal with their perceptions of what the job has been as well as management's perceptions of where they want it to go. You won't succeed unless you get these people on your side. Listening to them will help in this. Ask them what they would change if they could and what they woudl keep the same and why. Don't promise to make those changes, but seriously consider them in light of what management wants to change and see if you can make some of them.