I've got a few suggestions. Tiberias has a lot of old Roman ruins plus nice beaches at the same time.
Masada is another one that has historical value and outdoors activities at the same time, old fort from the Judean uprisings with a huge ramp that the Romans built up the side to let them get in and wreck the place.
Tsfat/Tzfat is a pretty cool place, it's where a lot of the spiritual hippy types hang out doing shrooms and singing for the messiah and shit. Interesting vibe.
Mitzpeh Ramon is a neat spot down south that looks like it should be an asteroid impact crater but actually isn't. It's surrounded by Kibbutzim so if you ever felt like living on a farm for a few months with little to no questions asked, you could do it here.
The Yad La-Shiryon tank museum in Latrun is a repository for a lot of the motley armored vehicles that have wound up in IDF service over the decades. You can see everything from indigenous Israeli designs to American, French, British, Soviet, and other flavors of surplus armored vehicles here.
I'll try to think of more shit.
Edit: more shit
If you're going to end up in the Old City of Jerusalem, which I don't recommend but whatever, one of the happening spots for tourism is Eir David. It's an area outside the walls that's got some fairly recent archeological work in it, and is 'new' in terms of stuff in the area that is officially opened to the public. You can also walk the walls themselves.
If you are feeling charitable in the Old City for some reason, there is an old senile woman named Luna who hangs about the arches between the Rova of the Jewish Quarter and the stairs going down to the Western Wall. She knows the area and can reliably get your money to actual needy people and keep you from getting duped by assholes who turn around and give it to families with 15 kids and no jobs to speak of. She was old as fuck the last time I was there so she may not be around anymore.
The Golan Heights are full of old abandoned military posts, a good one is Mount Bental, which has an old IDF outpost that was built on a captured Syrian outpost. It has a lot of intact bunkers and trenches, and good views of the surrounding Golan and Syria. Another good hike.
If you really wanna gay ops the IDF, there are a lot of unmarked border crossings with Jordan where a sufficiently hapless tourist can 'wander' into the wrong area and force a patrol to come by and pick you up or tell you to fuck off. If you get far enough you might even get sneeki breeki'ed by Sayeret Egoz, who use those same border crossings to do persistently gay recon ops in ghillie suits on the Jordanian side
Tel Aviv is a sufficiently libertine city to have a modern night life up to the standards of your average large American city, if that's your thing. Lots of homos, it's a bit of a fag capital of the Middle East, unironically. You can get weed here but enjoy paying your liver for some catgrass, you're better off getting hash from some of the jewish hippy types. I've never tried to get harder stuff. Alcohol is also obnoxiously expensive.
