I travelled to Hebron and Bethlehem last month.
Crossing the checkpoint was very easy. I just took an Israeli Egged bus to the settlement, visited the Cave of the Patriarchs, and walked to the checkpoint.
The Israeli guard didn't mind, just glanced at my passport and visa slip. A Palestinian hostel owner who happened to be on the street asked where I was staying. I told him I was going to stay with my friend Mo, who is a CouchSurfing host with hundreds of references, extremely hospitable, and truly legendary. You must visit him. That hostel owner said he knows Mo, and told me it's the right way!
I wasn't sure about how to take a bus and didn't want to pay for a taxi, so I walked for 45 minutes to Mo's place next to the university. During that time, people struck up conversations 3 times. That's more outgoing than the previous 3 days in Israel! People were happy to see a tourist there.
Water shortage is an issue. I was told that the Palestinian authorities can't improve the water supply because it's controlled by Israel. Palestinians pay double taxes, to both governments. Israeli shekels are used. It's peaceful, although the streets look like images from the news of bombed-out areas in Syria after 10 years of peace to rebuild and start businesses. What is more comfortable to you, riches with inequality or people living in equal poverty?
Israel, on the other hand, is rich and clean, with a good bus network (except on Fridays). People are more wary of strangers, though. I hadn't made an opinion of the political situation until my visit.
Re-entering Israel was easy, on a bus from Bethlehem to Jerusalem. Some military came on board and looked at the passports and visa slips, but didn't cross-check anything online.
On the way out at Ben Gurion airport, they didn't ask where I'd travelled on that trip. Instead, they noticed I'd been to Malaysia several years ago, and asked me many questions about that (Malaysia is a Muslim country). I had no problems with the Taiwan passport sticker [1]. They gave me another passport sticker [2] with a barcode beginning with "5", meaning I'm low-risk. They didn't check electronics. Most likely, you won't have a problem. If you're from a "Muslim country" then perhaps you can visit through the border with Jordan instead, just to avoid Israel. But Israel's quite nice to visit too (the sunset over the ocean from Haifa is beautiful).
[1] https://www.facebook.com/TaiwanPassportSticker/photos/a.1466...
[2] https://bemusedbackpacker.com/2017/01/16/what-to-expect-at-i...