Things to see/do: holy sites, Old City, David's Citadel, Mount of Olives, Mount Herzl, Israel Museum, Yad Vashem, are all but to name a few. Jerusalem also has a surprisingly good nightlife if you know where to go.
You can also easily get cheap coaches to the Dead Sea, Tel Aviv and other tourism hotspots.
I don't know about specific restaurant recommendations (also last time I gave specific recommendations my answer was removed as a violation). Generally speaking, you can't go wrong with popular shawarma, falafel and hummus places. Arabic food will almost always be good. You could also try various artisan Israeli restaurants, burger chains and European cuisine inspired places.