Taglit is just an introduction to Israel. There's so much more to see in the country. But a lot of ground is crammed into your ten day experience. When I went on Birthright as a participant, knowing I had an amazing trip planned for me in Israel, my planning for after the trip focused on getting out of the country.
You don’t need to be told that the Middle East is a volatile part of the world. You must always be aware of up to date situations on the ground and be responsible for your own safety. Israel has peace treaties with Jordan and Egypt which means the borders are open and you can travel by land to both. The West Bank? Off limits while on your Taglit trip, even to take the highway as a shortcut to the Negev dessert. Several parts are off-limits or restricted to Israelis. But it's certainly open to you. A visit to Lebanon and Syria are not happening. The borders with Israel are closed, traveling via Jordan wont work, and this is just not the time to go.
Jordan
Jordan remains the most stable neighbor of Israel’s—the safest to visit. The border has been open since the 1994 Israel-Jordan Peace Treaty. The most popular and spectacular site to see hands down is Petra, the ancient archaeological wonder. There's Wadi Rum, breathtaking desert landscape, the ancient capital city of Amman, Jerash, the largest Roman ruins in the Middle East, and the twisty Kings Way road. Sub-one hour flights between Tel Aviv and Amman can be had but are expensive. Most solo travelers cross the border from Eilat into Aqaba where a free visa can be obtained on arrival and Petra is not too far away. You can get between Jerusalem and Amman via the Allenby /King Hussein Bridge crossing, though the border has finicky opening times and can be very crowded, and is all around less pleasant. If visiting Petra is your intent, a circuit entering Jordan from Eilat and leaving through Amman, or vice versa, makes a lot of sense.
Egypt and the Sinai Peninsula
It may be best to split thoughts about the prospects for visiting the Islamic Republic of Egypt into two.
The Sinai Peninsula is a hop, skip, and a jump from Israel. You literally walk over from the Eilat border check point through a no-mans land and into Africa, Egypt, and the Sinai. The Peninsula, taken by Israel in the Six Day War and returned to Egypt in the Egypt-Israel peace treaty, is a place to laze the days away lying along or diving into the expansive Red Sea coast, looking across the Gulf at Saudi Arabia. The occasional Bedouin beach camp dots the coastal road heading south from the border to the land's southern tip and Sharm el-Sheikh. There are scattered low key beach resort options outside of Taba an Nuweiba. The 'backpacker's paradise' with a string of hostels and small hotels along a walkable strip, each with pillow strewn cafe/hookah bars fronting the water, remains Dahab. Anywhere you stay can organize tours into the interior, including a chance to take an overnight hike to the summit of Mt. Sinai, where Moses received the 10 commandments.
Then there is wondrous Egypt. Be aware of travel warnings and current events on the ground before deciding to visit as things are volatile. As of the summer of 2014, the situation seems stable enough that savvy travelers can think about returning. Stay away from gatherings or protests if they are occurring. That goes without saying for just about anywhere you visit. Cairo has the Pyramids of Giza and Sphinx, Antiquities Museum, and Coptic Cairo. Alexandria and Luxor can be reached from there if you have more time. There's a few ways to get from Israel to Cairo. Easiest would be to fly from Tel Aviv. There are direct flights four times a week. Mazada Tours operates a bus between Jerusalem/Tel Aviv and Cairo that takes about 13 hours, costs $110 round trip plus border taxes, and goes twice a week. It is possible to string the trip together on your own. It requires a bus to Eilat, a taxi or local bus to the border, and a walk through the border and into Taba. From there, East Delta buses make the trip two times a day, but it's difficult to confirm times which are not published online and could leave you waiting a while for a long ride across the dessert in an unconditioned coach. You need a visa in advance which can be obtained at home or from the embassy in Tel Aviv.
West Bank
There are plenty of reasons to visit the West Bank. Jews have settled or occupied (depending on your politics) several parts of this historic land of Judea and Samaria and Israelis are forbidden by their own government from visiting certain areas. They are all open to you. The bulk of foreign visitors visiting the Palestinian territories are surely Christian pilgrims visiting Bethlehem on organized tours. You can go on an organized tour too. Green Olive Tours has some great single and multiple day options.
Further Afield
Traveling further afield by air is possible. It takes less than an hour to land in Cyprus. Istanbul is a 2 hour flight and the competition with up to 7 daily flights can mean cheap fares if you want to visit Turkey. Eastern Europe is only 3 hours away if you want to get there. You're closer to Western Europe and Russia then you are at home if that's where you really want to get to after crossing the Atlantic. But remember, the terms of your ticket mean you'll need to return to Israel to be able to use your flight back to North America.
You don’t need to be told that the Middle East is a volatile part of the world. You must always be aware of up to date situations on the ground and be responsible for your own safety. Israel has peace treaties with Jordan and Egypt which means the borders are open and you can travel by land to both. The West Bank? Off limits while on your Taglit trip, even to take the highway as a shortcut to the Negev dessert. Several parts are off-limits or restricted to Israelis. But it's certainly open to you. A visit to Lebanon and Syria are not happening. The borders with Israel are closed, traveling via Jordan wont work, and this is just not the time to go.
Jordan
Jordan remains the most stable neighbor of Israel’s—the safest to visit. The border has been open since the 1994 Israel-Jordan Peace Treaty. The most popular and spectacular site to see hands down is Petra, the ancient archaeological wonder. There's Wadi Rum, breathtaking desert landscape, the ancient capital city of Amman, Jerash, the largest Roman ruins in the Middle East, and the twisty Kings Way road. Sub-one hour flights between Tel Aviv and Amman can be had but are expensive. Most solo travelers cross the border from Eilat into Aqaba where a free visa can be obtained on arrival and Petra is not too far away. You can get between Jerusalem and Amman via the Allenby /King Hussein Bridge crossing, though the border has finicky opening times and can be very crowded, and is all around less pleasant. If visiting Petra is your intent, a circuit entering Jordan from Eilat and leaving through Amman, or vice versa, makes a lot of sense.
Egypt and the Sinai Peninsula
It may be best to split thoughts about the prospects for visiting the Islamic Republic of Egypt into two.
The Sinai Peninsula is a hop, skip, and a jump from Israel. You literally walk over from the Eilat border check point through a no-mans land and into Africa, Egypt, and the Sinai. The Peninsula, taken by Israel in the Six Day War and returned to Egypt in the Egypt-Israel peace treaty, is a place to laze the days away lying along or diving into the expansive Red Sea coast, looking across the Gulf at Saudi Arabia. The occasional Bedouin beach camp dots the coastal road heading south from the border to the land's southern tip and Sharm el-Sheikh. There are scattered low key beach resort options outside of Taba an Nuweiba. The 'backpacker's paradise' with a string of hostels and small hotels along a walkable strip, each with pillow strewn cafe/hookah bars fronting the water, remains Dahab. Anywhere you stay can organize tours into the interior, including a chance to take an overnight hike to the summit of Mt. Sinai, where Moses received the 10 commandments.
Then there is wondrous Egypt. Be aware of travel warnings and current events on the ground before deciding to visit as things are volatile. As of the summer of 2014, the situation seems stable enough that savvy travelers can think about returning. Stay away from gatherings or protests if they are occurring. That goes without saying for just about anywhere you visit. Cairo has the Pyramids of Giza and Sphinx, Antiquities Museum, and Coptic Cairo. Alexandria and Luxor can be reached from there if you have more time. There's a few ways to get from Israel to Cairo. Easiest would be to fly from Tel Aviv. There are direct flights four times a week. Mazada Tours operates a bus between Jerusalem/Tel Aviv and Cairo that takes about 13 hours, costs $110 round trip plus border taxes, and goes twice a week. It is possible to string the trip together on your own. It requires a bus to Eilat, a taxi or local bus to the border, and a walk through the border and into Taba. From there, East Delta buses make the trip two times a day, but it's difficult to confirm times which are not published online and could leave you waiting a while for a long ride across the dessert in an unconditioned coach. You need a visa in advance which can be obtained at home or from the embassy in Tel Aviv.
West Bank
There are plenty of reasons to visit the West Bank. Jews have settled or occupied (depending on your politics) several parts of this historic land of Judea and Samaria and Israelis are forbidden by their own government from visiting certain areas. They are all open to you. The bulk of foreign visitors visiting the Palestinian territories are surely Christian pilgrims visiting Bethlehem on organized tours. You can go on an organized tour too. Green Olive Tours has some great single and multiple day options.
Further Afield
Traveling further afield by air is possible. It takes less than an hour to land in Cyprus. Istanbul is a 2 hour flight and the competition with up to 7 daily flights can mean cheap fares if you want to visit Turkey. Eastern Europe is only 3 hours away if you want to get there. You're closer to Western Europe and Russia then you are at home if that's where you really want to get to after crossing the Atlantic. But remember, the terms of your ticket mean you'll need to return to Israel to be able to use your flight back to North America.