Other than Turkmenistan which still has a rigorous visa program , Uzbekistan is the last of the Central Asian Stans to join in with having a relatively simple visa process for most countries.
Now most western countries are visa free, whether arriving by land or air, for 30 days.
USA passport holders, along with several others, do still require an evisa, which is very simple and can be found here .
The evisa takes 1-3 days to process and is valid for 90 days. So make sure you do it a week or so in advance to be safe, but not earlier than three months in advance. It will take around 15 minutes to complete the online form and you’ll pay online via the website. The only issue is that payment needs to be made with a visa card, so you just need to make sure that you have a visa card you can use.
Once the evisa is processed, it will be sent to you as an email attachment. Make sure you print this off and bring it with you to the airport or the land border.
If you are not eligible for the evisa, you will need to get a visa at an embassy, like everyone had to before. In this case you may need a Letter of Invitation which can arrange for you, or depending on your nationality, you might be able to do it without any supporting documents.
