I've used both SQL Express2005 and 2008 on vista with no problems. I think 2005 did show a compatibility warning when I installed it but it ran fine. I would try SQL Express 2008 just because its the latest version.
If you continue to get errors installing then I would check you have the latest windows service packs installed.
I've had a quick look at what could cause error 1603 and found this http://support.microsoft.com/kb/916760 . It sounds like it could be a bad installer, try downloading the installer again and see if you get the same problem. Also if your trying to install it from a network drive try copying the installer to your local PC and running it form there.