Okay, there's really nothing simple about joining a project as the PM when the project is already underway. You're going to need to jump right in and keep the project on track while you ramp up on everything about the project.
However, while you could probably write a small book or a long article on this topic, I have found that five is the magic number as far as the limit of "action items" or feedback points you can give someone and hope that they would retain it. Three is probably better, but I am hoping my readers are brighter than that. [;)
1 - Invest in Relationship Building
Take the time right up front to meet with all your team leads and key stakeholders one on one. Have an agenda going in, but try to keep the meeting informal and ask open ended questions to get people talking: ask each how they feel the project is going, what is working well, and what could be improved. You want to get several things out of this meeting - you need to quickly build rapport and find areas of common interest you can leverage to begin to build relationships with those you haven't worked with before; you want to see what areas are highlighted for improvement, so you can direct attention to them; perhaps most importantly, you want to try to sense how people are feeling about the previous PM no longer being there, which is probably why you would be joining after the project started. While the urge will be great to bury yourself in the project charter, scope statement, WBS, and project schedule, resist, RESIST!
2 - Get the Up to Speed, ASAP
In direct juxtaposition to the above, you will need to ramp up quickly on the project, but don't do it at the expense of rule 1 above. Ideally, you can get access to the project repository/library/whatever and gain an understanding of what the project is about, but if you have to do it after hours or on the weekend, it is an investment well made. I usually go for the project charter, scope documents, schedule, and risk register. You need to demonstrate competence and confidence to all the stakeholders you interact with in that first week or two, when everyone will be judging the new PM, and the team will be deciding if they are going to rally around you. As you review, make a list of questions you want to ask of team members and stakeholders as you meet with them.
3 - Trust the People on the Team
It may be the hardest thing to do sometimes, but you are going to have to trust the people already on the project to do the right thing. They have history on the team, they (should) know what they are doing, and you aren't going to succeed without them. More than this though, you need to show them that you trust them, by word and deed. In relationship to point 2 above, tell the team that you don't know everything, they are the project experts and you am going to rely on them. When an issue comes up or decision needs to be made facilitate the decision making process, but get the people with the project tribal-knowledge, so to speak, involved. Another thing that seems to work with most people is to ask them to explain aspects of the project they are the expert on.
4 - Do Something to Shake Things Up
I am not a psychologist, so I can't put this in proper technical terms, however I have found a significant, positive mental burst of energy and motivation can be had from the team by make some changes right away. It can be how meetings are run, how reports are presented, or even something like changing around how people are physically seated. Go for some low hanging fruit that you know will benefit delivery or the project. Certainly, you will be approaching the project with a fresh set of eyes, and you should try to bring enthusiasm with you to the assignment. Share your observations and ideas with the team in an engaging and motivated fashion.
5 - Resist the urge to blame the other guy (or gal)
I can't remember a single project where I have jumped in after it was started where people on the team didn't blamed issues on the previous project manager. Don't be that guy, it's shows a clear lack of accountability for the project you now are responsible for, and it is unprofessional. Be proactive and identify all the deliverables, action items, issues, etc. assigned to your predecessor and put them on your to-do list. Stay classy PMs!