Hospitality towards guests is an important part of Uzbek culture. If you are invited to an Uzbek home, you should take a small gift for your hosts. A souvenir item, like a picture book, from your home country is ideal, but if you have brought nothing suitable with you a gift such as a bouquet of flowers (get an odd number) or box of sweets is fine. Note that an invitation to go to someone's house for 'a cup of tea invariably means something more substantial: often a full meal. If you are visiting a household in which lood may be in short supply, take at least as much as you expect to receive, and press thai upon your hosts as a gift. Even if they initially refuse to accept it, persevere.
On entering a home in Uzbekistan (true for an ethnic Russian family as well as a Uzbek one, and for a yurt as much as for a house) you should remove your shoes at the door. There is usually a mixture of assorted slippers and flip-flops available to wear around the house. You should not shake hands or kiss across a doorway or step on the threshold as this is believed to bring bad luck.
In more traditional households you may find that men and women are entertained separately. Although there is little concern about women being seen by non-family members, they often feel more comfortable sitting and talking with their own sex. Foreign women frequently have the fortunate position of being able to participate in both male and female social interactions, and so can get to know all parts of the family with greater intimacy.
At the end of the meal, thanks are given by the act of bringing the hands together in front of the face, then moving them down in an action symbolising a washing gesture. This is the signal for everyone to get up from the table. You shouldn't continue to pick at food after this point.
Mosques, shrines and other holy places often have their own sets of rules and you should endeavour to observe these. If in doubt, ask, as it is better to appear naive than disrespectful. Requirements are likely to include removing your shoes, covering your head (women only) and wearing long trousers or a skirt that covers the knees. Shorts and strapless T-shirts will get you strange looks anywhere in Uzbekistan and at religious sites they really are unacceptable. You may also be expected to wash your hands and face and in some places only one gender is permitted to enter. Whatever your personal opinion, you should respect the community's wishes.