"Today is April 1st."
Translation:Bugün nisan bir.
yes, birinci. But we don't use ordinal numbers (birinci) when we talk about dates
That would have been my choice of wording as well. Without the "dır", though. You don't need it in this context. "Bugün 1 nisan" is accepted.
Do people in Turkey 'celebrate' April 1st? Or is there something similar? :)
Not really, but it is becoming more and more known.