Current Time of Timezone

The time zone IDs you've given are the ones from the IANA time zone database, aka TZDB (aka Olson, aka tz). They're not supported by .NET (although .NET Core running on Linux/Mac will probably do what you want).

My Noda Time project does support them though. You'd use:

var zoneId = "America/Antigua";
var zone = DateTimeZoneProviders.Tzdb[zoneId];
var now = SystemClock.Instance.GetCurrentInstant();
var zoned = now.InZone(zone);
Console.WriteLine(zoned);

That uses the SystemClock explicitly - in real code I'd advise you to accept an IClock via dependency injection for testability - inject SystemClock.Instance when running the application, but use FakeClock for testing.

Alternative equivalent code demonstrating ZonedClock:

var zoneId = "America/Antigua";
var zone = DateTimeZoneProviders.Tzdb[zoneId];
var systemClock = SystemClock.Instance;
var zonedClock = systemClock.InZone(zone);
var zoned = zonedClock.GetCurrentZonedDateTime();
Console.WriteLine(zoned);