Hi Marcel 2586,
Your goal should be "domain trust" which is the default. That provides the least cost option since nothing needs to be installed and most features are supported. Restricted Trust mode is good for forms sent in email but is fairly limited. Full Trust is used for forms that have code which access files on a local computer or run other applications. Full Trust is not preferred because a) it reduces security, and b) is more costly to install (you have to either give people an MSI to install or sign the XSN with a code-signing certificate which can be tricky for your users to handle). Domain Trust is the "nakamichi" or middle road which all good monks recommend, and of course we know InfoPath forms development can sometimes feel like you're at the monastery ... (just kidding)...
Previewing your form locally is a special form of trust - it exists between Restricted and Domain modes. This special mode isn't available for published form and you wouldn't want it anyway since it's more restricted than Domain nd domain-trust is the best tradeoff.
My guess is that your form has some code in it (to query your Access DB?) which is preventing you from using the default mode.
If so, try signing the form with a locally generated certificate and publishing it as Full Trust. You may have to accept a certificate when opening it and that's the main hassle your users will experience. The other issue is that InfoPath 2010 only generates personal certificates good for one month (2007 used to do multi-year ones). At any rate, if you need to go Full Trust, you'll need to create a domain-trusted code-signing certificate using your intranet's certficate authority service or by purchasing one from an internet vendor (GoDaddy, Verisign). It's not too costly to do the latter but the main cost is showing your users how to jump through the hoops to accept the certificate (even though this isn't really that bad).
Hope that helps a wee bit...
Gyatei gyatei hara gyatee bouji sowaka ...