When I arrived in Bukara at 6 a.m., traveling by overnight train from Samarkand, it took my breath away! People had been talking constantly to me about Bukhara over the last two weeks of my visit, and I can confirm that the place isn't overrated at all - it genuinely is that beautiful.
To my eyes, there are two trips that you absolutely must take:
A little bonus: the Modrai Khan and Abdullah Khan madrassas, which are a little way away from the center, are superb and are forgotten by the majority of tourists.
One can have a somewhat superficial image of Uzbekistan with its "museum" cities. Personally I really liked Bukhara, because we came in contact with the locals. I especially could meet some Uzbeks thanks to the Couchsurfing.com website.
These encounters allowed me to have a local's view of the city while avoiding the traditional tourist destinations. I went to lose myself in the narrow streets of Bukhara, and I let myself be guided by the city's minarets, including the Kalyan minaret, impressive at 48 meters tall.
For music lovers, I recommend that you discover Uzbek folklore by going to a traditional music concert in an ancient madrassa. Nadir Divan-begi, for example.
Don't forget to visit the city's bazaars to soak up the local atmosphere.
Avoid visiting Bukhara in full summer or winter, because this city sees significant changes in temperature. Nevertheless, the city has a clear blue sky most of the time.