6 weeks should give you plenty of time to visit the parts that are open to tourists.
Things that should not be missed:
- Sana'a - the old city, qat chew, weddings, the brand new president's mosque, KFC garden on a weekend evening, not for the chicken, but to watch Yemeni families relax and enjoy the "outdoors", lunch at the fish market, Bait Baws, Wadi Dhahr, Shibam, Kawkaban, Thila, Hababa, etc. Recommended hotels for low-mid range budget: Arabia Felix and Dawood, for high-end: Burj Al Salam. The weather is warm and sunny in the day and cool at night, absolutely perfect for traveling.
- Haraz mountains: enough trekking for a week.
- Al Mahwit and Al Tawila: great scenery.
- Ibb and Jiblah
- Hadhramaut
Things should be seen if you have time:
- Taiz, Aden
- Zabid, Bait Al Faqih(the Friday market)
- Socotra Island, Kamaran Island
The war is raging in the North, people are rioting in the South, and the tribes are restless in the desert. Many areas are completely closed to tourism, such as Saada, Shihara, Marib (might be doable with an expensive guided tour), and the area between Aden and Mukalla. It certainly isn't the best time to visit Yemen, but if you are determined, dress like a local, use local transportation, and stay away from other tourists. Yes, one foreigner is a much smaller target to kidnappers than a group.
Feel free to ask if you more specific question.