below are some interesting historical locations in Syria:

Damascus: the oldest continuously inhabited city in the world.

The village of Ma'alula: one of the few places in the world where Aramaic, the language of Jesus Christ, is still spoken for the last 3000 years. It is 50 km from Damascus (to the north west). You will find yourself in legend and historic place. You can visit Seidnaya also it is ancient village as well and near by Ma’alula.

Ugarit: Nowadays called Ras Shamra, 16 km to the north of Lattakia (you can spare one night in Latakia, Ugarit, the kingdom that had a golden past in administration, education, diplomacy, law, religion and economics between 16th and 13th centuries BC.
First alphabet in the world “the nail style alphabet” was founded in Ugarit where Latakia Syrian City is placed today.

Palmyra (Tadmor): Palmyra is in the heart of Syrian Desert. is located near a hot-water spring called Afqa which make it an ideal halt for caravans moving between Iraq and Al-Sham (present day Syria, Lebanon, Holy Land and Jordan), trading in silk from China to the Mediterranean. This strategic location made Palmyra prosper in a well-established kingdom from the 2nd century BC.

Bosra: is an extremely ancient city mentioned in lists of Tutmose III and Akhenatcn in the fourteenth century BC.

Amrit: The ancient city of Amrit is only 7Km far to the south Tartous, it was established and inhabited by the Amorites at the end of 30th century and the beginning of the 20th century B.

Aleppo castle: The citadel's fortified entrance is a marvelous example of Arab military architecture. It is located in the old city of Allepo.
And many many other nice places to visit.

Following are some of nice natural, green mountains and vallies:
Ras Al Bassit: Located on the northern part of the Syrian shore, about 40 km to the north of Latakia, it is one of the most beautiful spots on the Mediterranean: a wide bay, with clear water and clean sand, surrounded by mountains and green hills.

Banias: located at the foot of the hill of Qalaat el-Marquab, 55 km to the south of Latakia and 35 km north of Tartus.

Finally, it is better if you find traveling agent and program for your short one week tour. This will allow you to see most of the beautiful places there and will limited the expense there.