How to concatenate certain strings in PowerShell?

You can use .tolower() and .Substring() methods to do this.

Import-Module -Name ActiveDirectory
$username = Read-Host -Prompt ('Please Enter the Username')
$startdate = Read-Host -Prompt ("Please Enter the user's Start Date (Format: Feb 2022 = 022022)")
$user = Get-ADUser -Identity $username
$finalvariable = $user.GivenName.Substring(0,1) + $user.Surname.Substring(0,1).tolower() + $startdate
$finalvariable

You could use this as a reusuable function with arguments like below.

Nicknamer

function Generate-NewID {
  <#
    .SYNOPSIS
    Nicknamer

    .DESCRIPTION
    Provides nickname given username and date.

    .PARAMETER user
    
    The samAccountName of the User.

       The specified date.

    .PARAMETER date
    
      The specified date.

    .EXAMPLE

      Generate-NewID -user mjones -date 010122
    
        > Mj010122

    .NOTES
    Place additional notes here.


    .OUTPUTS
    Nickname in the correct format.
  

#>



  param
  (
    [Parameter(Mandatory = $true,
      ValueFromPipeline = $true,
      HelpMessage = 'Please Enter the Username')]
    [Object]$user, 

    [Parameter(Mandatory = $true,
      ValueFromPipeline = $true,
      HelpMessage = "Please Enter the user's Start Date (Format: Feb 2022 = 022022)")]
    [Object]$date
  )
  Import-Module -Name ActiveDirectory

  try {

    ## test if AD user exists
    $null = Get-ADUser -Identity $user
  }
  catch [Microsoft.ActiveDirectory.Management.ADIdentityNotFoundException] {
    ##can't find that user
    Write-Warning -Message 'There was an error finding that user.'
  }
  catch {
    ## a different issue 
    ## if you are seeing this you need to remove the
    ## "$null =" in the above try{} block
    Write-Warning -Message 'Other issues...'
  }
  finally {

    $inputuser = Get-ADUser -Identity $user
    $startdate = $date
    $output = $inputuser.GivenName.Substring(0, 1) + $inputuser.Surname.Substring(0, 1).tolower() + $startdate
    $inputuser = $null
    $user = $null
  }
  $output 
}