Analyse XML file with powershell script

The statements you've posted are completely independent at the moment - the first one outputs the textual encoding of all the transactions nodes, and the second and third ones simply do nothing, because $_ no longer has a value assigned to it at that point.

To properly "connect" them, you'd have to either place the filtering logic inside the first ForEach-Object block, eg.:

Select-Xml ... |ForEach-Object {
  if($_.Node.Response -eq 'Error'){ $_.Messages }
}

... or store the output from each step in an interim variable, eg.:

$allTransactions = Select-Xml ... -XPath '//Transaction' 

$allTransactions |ForEach-Object {
  if($_.Node.Response -eq 'Error'){ $_.Messages }
}

I should point out that ForEach-Object { if(...){ $_ } } is a bit of an anti-pattern unless your code has more complicated side effects - the more idiomatic solution would be to invoke the Where-Object cmdlet to filter the output from Select-Xml:

$allTransactions |Where-Object {
  $_.Node.Response -eq 'Error'
} |ForEach-Object Messages

While these suggestions might solve your problem, I strongly recommend not doing any of that - XPath is much more capable than what you're currently using it for :)

How do I get only get the transactions where the Response is "Error"?

I'd suggest simplifying your code by using a more accurate XPath expression with Select-Xml - one that looks for exactly what you want:

Select-Xml -Path C:\Users\user\path\path\file.xml -XPath '/Paths/Task/Transaction[Response = "Error"]'

Is there a possibility to just have the first message line of each "Error" transaction as a output? So that I have a list of all the "Personal-Nr" that were in an error transaction?

Sure thing!

Once again the easiest way is to modify the XPath expression, this time to only resolve the first <Message> node under a <Transaction> fitting the criteria above:

# beware that index selectors in XPath start at 1, not 0
//Transaction[Response = "Warning"]/Messages/Message[1]

But that's not all! XPath has several useful functions - so we can go one step deeper and have XPath extract and decode the message text for us too!

//Transaction[Response = "Warning"]/Messages/Message[1]/text()

This will cause Select-Xml to return a node set consisting of XmlText instances which you can convert directly to strings to get the raw string content.

Putting it back together with Select-Xml, you end up with something like this:

$filePath = 'C:\Users\user\path\path\file.xml'
$xPath = '//Transaction[Response = "Warning"]/Messages/Message[1]/text()'
$messages = Select-Xml -Path $filePath -XPath $xPath |ForEach-Object ToString

$messages