Here are a few resources that I have personally found really helpful in trying to live out Acts 17:

First, I have often given friends, clients and family the audio Who is This Jesus?
(it is free to download – 1st on on the page).  I put in on a CD for
them and ask them to listen to it and offer to go to coffee or lunch to
discuss it.  It is about the historical evidence for the resurrection. 
The reality of the resurrection of Jesus was what the apostles
consistently gave as the reason why we ought to believe the Gospel.

Secondly, if they have particular objections and are readers I give then Tim Keller’s The Reason for God
(and once again set up a time to meet to discuss it).  I have given
this to clients, collegues and family that are readers.  For those who
are not likely to read a book you can download them messages (that have
similar content) from The Reason for God website.

Now, to tune yourself up to engage people with the Gospel I would highly recommend two books:

The Heart of Evangelism,
Jerram Barrs – This books is great for teaching you how to build
bridges, show respect and truly understand those you are called to
reach (very much in the spirit of Acts 17).  If you’re not a reader get
the same content as a podcast here.

The Reason for God,
Tim Keller – This book will help you learn how to answer objections to
the Christian faith in a consistently Gospel-centered way (in the first
half) and how to explain the difference between the Gospel and religon
(in the second half).  The second half will minister deeply to your own
soul.  If you’re not a reader there is an audio book version. 

The
best preparation for sharing Christ is being electrified by the Gospel
in your own heart (I Peter 3:15a) and for that you can’t do better
than: Centrality of the Gospel (free PDF), The Cross of Christ (John Stott), and Transforming Grace (Jerry Bridges).

He promised to go with you, live the mission!