I would travel in an organised group at the time around Easter. I would like to do some sightseeing, but wouldn't insist on anything even remotely dangerous. I'm not sure about the arrangement yet, but I guess that the trip will be localized in Jerusalem, i.e. I don't expect to visit other cities.
I can make no subjective comment as could one who lives in the region. If you are a US Citizen, the United States Department of State, Bureau of Diplomatic Security, Overseas Security Advisory Council (what a mouthful) published, in April of 2015, the Jerusalem 2015 Crime and Safety Report which covers everything from what it calls terrorism to petty crime. This report is, of course, aimed for the US Citizen traveling to Israel, Jerusalem and the Gaza strip.
A non-government page to visit for example, might be Travel to Israel - Is it Safe?. Even though it is created and run by a single individual, the guidance is helpful, appears to be from someone who actually lives in Jerusalem, and is not all positive (or negative).
Carefully reading through both sites, IMO, you're as safe (or not) in Jerusalem as anywhere else in the world, barring of course, actual war zones.
If you stick to being an average tourist, Jerusalem is as safe as any large city. Do be aware that in Israel you have to pass security checks at every train station, mall and at many tourist places. Though that actually makes stuff safer, so don't worry about that.
If you intend to visit places like Mea Shearim, you are advised to dress properly. That means - especially as a woman - : long sleeves, no exposed body surfaces etc.
I can't say with certainty, but Eastern Jerusalem is probably considerably unsafer than Western Jerusalem.
Disclaimer: my partner is Israeli. I'm a Dutch Caucasian. I've been to Jerusalem, and I never experienced any issues.
Having lived in Jerusalem for 10+ years, and coming from Johannesburg (having lived in Johannesburg since I was born), I can safely say that Travel Advisories, however helpful, are not going to tell you geo-specific safety, since a lot depends on context, timing, and location.
I would, for example, not recommend young ladies to walk out at night late (say 2 am) in any part of Israel alone, especially in the city center of Jerusalem. That being said, it is cautionary, I would feel comfortable for a young man to do the same in the suburbs, but not the city center.
I wouldn't dare do that in the suburbs of Johannesburg, not at 8 pm, not at 3 am, ever.
That being said. Israel is much safer than you think. In terms of terrorism, there are hotspots, and mainly Jews are the targets (although there have been kidnappings of foreigners - this is a rarity but it does happen - more about this below).
The Old City is a hotspot, the Jewish Quarter is relatively safe and there are army personnel everywhere. The Muslim Quarter and Christian Quarters are no-go areas for Jews in general, however, the Arab Market is frequented by Jews who walk to the Western (Wailing) Wall on Jewish Holidays and the Sabbath.
The Via Dolorosa covers both the Muslim and Christian Quarters and one should be aware of where you are in this area. It is safe (for tourists) but you should be aware of your surroundings. The Muslim Quarter sometimes has riots and protests. They usually start on the Temple Mount and spill over into the Muslim and Christian Quarters.
That being said Eastern Jerusalem is mostly Arab - Arab Christians and Arab Muslims and Druze all inhabit this area - which borders Archaeological sites (like King David's old city).
Western Jerusalem is Jewish. Kikar Safra (City Hall) and the Municipal compound (at the intersection of Yaffa st. and Shlomo HaMelekh St.) is where the Jerusalem Municipality is housed (in different buildings). This is a no-go zone - you don't need to be there, and if you do, move quickly to where you need to be. There have been instances where some incidents have occurred.
The Kennesset Building (on Rupin St.) is heavily guarded.
Fridays are intense periods. Being in the Muslim Quarter on a Friday is not a good idea. On Sundown Friday, is an intense time for Jews preparing for the Sabbath and tensions run high, avoid markets and the like on Friday Afternoon from about 2 pm onwards.
Jewish holidays also begin at Sundown the previous day, make sure to be aware of these, since they're confusing to the uninitiated. The city shuts down mostly on Jewish Holidays and the Sabbath (Saturday). That is you won't get bus service and many taxi companies don't operate late afternoon on Friday until about an hour and a half after sunset on Saturday - the same is true for Jewish holydays - the evening before until nightfall the following day.
In the dominant Western culture, independence and creativity are respected and garner mass appeal. Certain vulgarities are not only allowed; but in some circles encouraged. Religious neighborhoods (Jewish, Muslim and Christian alike) require certain decorum and respectability. Consider: would you walk into the Vatican as a Catholic in a tank-top and shorts? Or would you wear something more respectable and formal, especially if you were meeting the pope or president? Religious sects require the same sense of respectability. Please refrain from wearing revealing outfits in Jewish, Muslim or Christian religious neighborhoods. Women should wear skirts that cover the knees, sleeves that cover the elbows, the neckline should be covered, and revealing clothing should be avoided in those areas - out of respect. Those communities consider anything less as public indecency and find it very offensive. Men should wear button-down shirts and pants/trousers that go down to the feet - no shorts. Sandals are a no-no here, not just because its disrespectful, your feet are going to get hurt in the rough and tumble of dirt, sidewalk construction and lots of people walking around. Consider how you would dress in front of a president of your country, and follow that lead. If you do not, you might be pelted or stoned, or asked to leave the area.
Jewish religious areas include the Wailing Wall and its surrounds (incl. the Jewish Quarter), Mea Shearim / Geula district and Beis Yisroel. There are many other religious communities, but are not really tourist areas.
Overall Jerusalem is a very safe place, as long as you keep your wits about you, avoid hotspots during certain times and days. Most tourists come to Jerusalem and have an awesome time without incident.
Kidnappings - These occur very infrequently. Tourists should not visit the Jerusalem Forest unless in a guided tour. The Jerusalem Forest is huge, and there are areas that are lovely to walk in, but are simply not safe. Only go there in a group with a guide and do not wander off on your own under any circumstances - use the lavatory beforehand.
Safety is more than just textbook terrorism. If you look like a tourist flashing dollar bills and cameras at every pedestrian crossing will make you an easy target for pick-pocketing and the like - try to dress normally and do not flash money wherever you go. Don't be too loud and conspicuous. Try to blend in.
Jerusalem is a hot city. Israel is a hot, middle-eastern country. While some areas are milder than others, it can be deceptively cool. Make sure to drink at least 1.5 liters (0.4 gallons) per day if you can. I would not drink tap water in any foreign country I'm not familiar with as a precaution.
More people in Israel die from dehydration and road accidents than terrorism. So make sure you keep an eye on the road, only cross at crossings and when the lights are green. Keep drinking fluids and make sure you have medical coverage. I would highly recommend opening up a tourist medical aid package which are relatively inexpensive and cover you for even routine doctor's appointments and basic medicines. Many tourists get travel diarrhoea, which can be dangerous if not looked after properly - being in a foreign country, without sufficient language skills can be a dangerous combination. Talk to your doctor about what precautions you should take.
Don't forget to enjoy the place, and absorb the air!