"What are the responsibilities and duties of a successor trustee of an existing trust?"
As trustee you have an obligation to invest trust assets, communicate (although the degree to which you must will depend on the trust and state law) with beneficiaries, carry out the terms of the trust, maintain records of all trust activities, etc. You will likely have to file income tax returns for the trust (unless its characterized as a grantor trust). You often can hire, at the expense of the trust, an investment manager to invest trust monies (but depending on the trust and state law you may still have some liability for investment decisions), an accountant to prepare income tax returns, and an attorney to advise you. To what degree you need this help will depend on the trust agreement, the assets the trust holds, your expertise, etc.
State law (prudent investor act, principal and income act, etc.) will also govern what your responsibilities are and how you must carry them out.
Your question is a great one, but very broad so tough to answer in more detail. However, you off to a good start since you need to understand your responsibilities. Being a trustee is a significant responsibility to take on, and if you don't handle it properly can expose you to personal liability.
There are a number of similar Q&As and some relating planning tips. Look around. Also, a recent issue of our Newsletter "Practical Planner" had a detailed checklist on steps to take for an irrevocable life insurance trust. Many of the steps will be applicable to other irrevocable trusts as well. You can find this newsletter on www.shenkmanlaw.com in the newsletter tab.
The book "The Complete Book of Trusts" by Martin Shenkman (you can click through to it from the book section of www.shenkmanlaw.com) has a lot of information on this topic.