Powershell Script Question - Beginner

How would I go about creating a Powershell script that says: if a user is a member of one AD group and their machine is in another AD group run this bat file?

Well, firstly, I would recommend getting the Quest AD Cmdlets.  They make scripting for AD a whole heap easier.

This is untested as our computers aren't members of any groups but it should work. 

 
 $UserGroups = (Get-QADUser <username>).memberof 
 $CompGroups = (Get-QADComputer <computername>).memberof 
 If ($UserGroups -Contains "X" -And $CompGroups -contains "Y") { 
    Invoke-Expression "batfile" 
     } 
 

And if you really MUST have it as a one-liner

 
 If ((Get-QADUser <username>).memberof -Contains "X" -And (Get-QADComputer <computername>).memberof -contains "Y") {Invoke-Expression "batfile"} 
 

The QAD cmdlets are handy. Here is how you would do it without them:

$domain = [adsi]''
$searcher = New-Object System.DirectoryServices.DirectorySearcher
$searcher.SearchRoot = $domain
$searcher.filter = "(&(ObjectClass=User)(name=<USERNAME>))"
$UserGroups = $searcher.FindOne().properties.memberof
$searcher.filter = "(&(ObjectClass=Computer)(name=<COMPUTERNAME>))"
$CompGroups = ($searcher.FindOne().properties.memberof
 If ($UserGroups -Contains "X" -And $CompGroups -contains "Y") {
    Invoke-Expression "batfile"
     }

Get an R2 box for mgmt, and install AD web service on your DCs.  New R2 cmdlets have all sorts of great AD stuff, and group policy cmdlets as well.