I'm planning a trip by bus or train, even a shared taxi if necessary (but I prefer train), from Uzbekistan to Kyrgyzstan, but I don't plan to get a Kazakhstan visa, so better not to go through Kazakhstan. Is it possible?
Yes it's possible. There are three main border crossings between the two:
Dostyk/Dustlyk: main border crossing on the road between Osh and Andijon. Public transport and shared taxis are available on both sides. Open from 7am to 7pm.
Khanabad: border crossing on the road between Jalal-Abad and Andijon. Update: may 2013, jalalabad- khanabat border was closed.
Uch-Kurgan: border crossing on the road between Karakol and Namangan. It’s definitely open. Guards on both sides are generally friendly. It’s a very small crossing, and there is no public transport going there. Will take 1-2 hours to pass through.
As someone who has crossed from Uzbekistan into Tajikstan, it's pretty much impossible to get a shared taxi all the way. You generally have to get one from Tashkent to the border, or to the town at the border and then a small one to the border.
You then walk (which in July/August sucks!) in the heat the long walk between the two posts, and on the other side get surrounded by excitable taxi touts who want to charge you tons. Pause, take a sip of water, and then start negotiating :) They'll take you to the next town you want to get too ;)