I'm an Israeli (not natively born, but lived there for 20 years) living now in Oz for the last 7 years. Have just come back from a business + family trip in Israel.
As others indicated, you will be perfectly safe there while applying some common sense. It's safer to go around with locals, but even on your own you should be fine. Most people understand and speak reasonable English. Watch for pickpockets in areas like markets, Jerusalem/Jaffa old city etc.
Getting around in Tel Aviv area is not difficult, as the public transport is quite good and taxis are abundant (you can also use Gett and Uber) and not as expensive as here. On the other hand, car hire is very expensive (practically more then anywhere else), and the quality of hired cars/service is quite poor. The fuel is expensive too. And the most difficult part is driving. Not as bad as India or Thailand, but pretty scary by common Western standards. I'd say don't even think of hiring a car unless you want to explore more distant/rural places on your own. Otherwise, the intercity busses/trains would serve you just fine. Do take into account that most public transport doesn't operate from Fri afternoon till Sat night, so you'd probably prefer to stay in the city during that time.
As for places to visit, there are so many! Jerusalem, Jaffa, Haifa, Acre, just to name a few. For scenic views and breathtaking nature, consider the Dead Sea, Mitzpeh Ramon, the Golan Heights. There are also beautiful caves in the Judea Desert and Rosh Ha-Nirka. And, as others indicated, Tel Aviv is a great cosmopolitan city, with plenty of night life, bars, restaurants etc.
The Mediterranean is cold Nov-Apr (not by Melbourne standards though), the swimming season is May-Oct. The beaches in Tel Aviv, Netanya, Haifa and Acre are great.
Last but not least, you can go to Eilat. It's a resort city on the Red Sea. Pity it takes 4-5 hours to get there by bus. Be prepared for 40+ degrees there during Jun-Sep.
Any other question, drop me a note!