I'm also a Christian and have been to Palestine/Israel 7 times for a total of about 2 years (trips ranged from weeks to a year).
There is a Christian Information Center located near Jaffa Gate, across from the Tower of David. Their website is http://198.62.75.1/www1/ofm/cic/CICmainin.htm and there
they list Christian guesthouses. I lived in a West Bank village but when in Jerusalem, I've stayed at the Knights Palace a few times. It's in the Old City, close to New Gate and also not far from Jaffa Gate. Their website is http://www.knightspalace.com It's run by the Latin Patriarchate (Catholic) Church.
In the Old City, visit the sites most important to each of the faiths - the Holy Sepulchre (Christian), the Dome of the Rock (Muslim), and the Wailing Wall (Jew). The Old City is made up of 4 quarters so visit each: Christian, Muslim, Jewish, and Armenian. Walk on top of the Old City wall for some nice views. Also visit the Garden Tomb, which is not far outside the Old City, by Damascus Gate.
If you'd prefer to go with a tour group rather than explore on your own, then contact the Christian Info Center to see if they have list of guides.
Do NOT take an Israeli operated tour company. As a Christian you'll want to see some of the sites within the West Bank (Palestine) and also learn more about Arab Christians there so be sure to ask for an Arab-run company!
Of course you'll want to go to Bethlehem, which is only about 5 km south of Jerusalem but a major checkpoint to go through between them. Also, the Dead Sea is amazing so spend some time floating on top of the water!
I really encourage you to get a tour group that visits a Palestinian city or village, and preferably one that includes meeting with local people. Ramallah is about 1/2 hour north of Jerusalem and a city. Within 15-20 minutes of there is the village of Taybeh (about 99% Christian) and Bir Zeit (about 60% Christian). Both beautiful villages worth visiting.
Be prepared to be surprised! What you read/see in the media is not all that is going on 'on the ground'. What the media often omits is the extent of the oppression and mistreatment of the Palestinians and what they have to live through. There's enough that speak English that you'll easily find someone to talk to. Ask questions - I've travelled a lot throughout the world and I must say the Palestinians are amongst the warmest, most helpful people I've ever encountered.
I haven't taken a tour myself but just did a search and found a couple of companies:
http://www.guidingstarltd.com/
http://www.awad-tours.com/
Probably best to contact the Christian Info Center for recommendations on Palestinian tour companies to get a longer list to consider.
If you have any other questions/concerns, feel free to email me. Have a super trip!