How to check which Azure Active Directory Groups I am currently in?

Here's a few different ways other than using the Portal

Azure AD PowerShell Module

Get-AzureADUserMembership

$user = "user@domain.com"

Connect-AzureAD
Get-AzureADUserMembership -ObjectId $user | Select DisplayName, ObjectId

Microsoft Graph PowerShell Module

Get-MgUserMemberOf

$user = "user@domain.com"

Connect-MgGraph
(Get-MgUserMemberOf -UserId $user).AdditionalProperties | Where-Object {$_."@odata.type" -ne "#microsoft.graph.directoryRole"} | ForEach-Object {$_.displayName}

Microsoft Graph API HTTP Request through PowerShell

List memberOf

$user = "user@domain.com"

$params = @{
    Headers = @{ Authorization = "Bearer $access_token" }
    Uri     = "https://graph.microsoft.com/v1.0/users/$user/memberOf"
    Method  = "GET"
}

$result = Invoke-RestMethod @params
$result.Value | Where-Object {$_."@odata.type" -ne "#microsoft.graph.directoryRole"} | Select DisplayName, Id

Microsoft Graph Explorer

GET https://graph.microsoft.com/v1.0/users/user@domain.com/memberOf

For Portal, simply click on the user for which you want to find this detail and then click on "Groups" button.

If you want to use PowerShell, the Cmdlet you would want to use is Get-AzureADUserMembership.