Israel for 10 days
 

I have 10 whole days in Israel. Flying into and out of Tel Aviv. I'd like to see as much of the country as possible. I'm also interested to know if anyone has gone to Beirut and a city in Jordan. I love nature, history and good food.

What do you suggest?

I recommend narrowing your focus; 10 days is not enough for all of Israel, Beirut, and Jordan.
With 10 days, you might consider 2 days in Tel Aviv, 4 days in the North, 4 days in Jerusalem, including a day trip to Masada, Qumran, and Ein Gedi.

I just did two weeks in Israel. 3 Nights in Tel Aviv and 10 in Jerusalem. We took three days/two nights from Jerusalem and did a tour with Abraham Tours to Petra/Jerash/WadiRum which was fantastic and the highlight of our trip really. Depending upon the reason/focus of your trip you could fit in Jordan IMO.
From Jerusalem we did day trips, with a driver and guide, to Bethlehem and Masada/Dead Sea another day. We did not get to the north, which I wish we had.The rest of the time was spent in Jerusalem.
You can transfer from Jerusalem to the airport no problem, so start in Tel Aviv and end in Jerusalem.

I think Abram has basically the correct idea. Which city would you want to visit in Jordan. Depending on the city it can take 2-3 days from your Israel trip and assuming you are gong AFTER Jan 1, 2016, 2 of the 3 crossing require you to obtain your VISA in advance.

You can not enter Lebanon after having been in Israel, so that is not happening.

Thank you everyone for all your helpful information.

So here is what we are thinking as of now:

Jerusalem 28-31 of December
-Visit Dead Sea/Qumran/Ein Gedi/Masada

Tel Aviv 31 Dec.-2 January

Haifa 2-3 January
-Acre

Nazareth 3-5 January
-Sea of Galilee
-Cesar

After this we still have 2 days and we are really shocked by the cost of the entry/exit fee to get into Jordan. We were originally thinking to spend the last few days in Petra but now we are not sure. Does anyone recommend anything either way?

There's plenty to do in Jerusalem to occupy more than 3 days; I'd add at least one more day there.
The North also has lots to do--you could easily add more time there.

Are you planning to rent a car? If yes, pick up the car in Jerusalem, drive north to the Galilee, then drive to Tel-Aviv via Akko and Haifa. Return the car when you arrive in Tel-Aviv and spend the last few days of your vacation there.

The visa is less if you go to Jordan for two nights instead of one.

Still too many destinations for the short period of time.

I strongly recommend Petra and Wadi Rum - absolutely spectacular. It would be a shame to miss Petra if you are in the area. You need a full day for Petra, so two overnight stays in Petra.

For my taste, Tel Aviv and Haifa are second-rank destinations, also Caesarea (at least if you know Roman archeology from other parts of the world). Jerusalem deserves some time and you need at least a full day for En Gedi and Masada. There is not much to see in Qumran (the scrolls are in the museum in Jerusalem), but swimming in the Dead Sea is a once-in-a-lifetime experience.

I would really try to focus, like many others have already said. I recently went to Israel for my 2nd trip and I still feel like I have a lot more to explore in that beautiful country.
Jerusalem alone can easily take 4-5 days of your trip if you're an history fan. An anecdote on Jerusalem from my recent trip - there are several lovely new hostels in the city that proved to be relatively cheap and convenient - we enjoyed The Post Hostel the most, but there were others that we liked as well.
I recommend to also try and see Petra. It's quite simple to go there from the city of Eilat (which is what we did) and it's definitely worth the trouble (it's magnificent).