Welcome

How do you write Welcome in the Japanese language?

Hmm, if only Google had a simple translation page...

ようこそ (yokoso)
ようこそ 日本 へ (yokoso Nihon e; welcome to Japan)
Here is "YOKOSO! JAPAN" by the Tourism Agency in the Ministry of LITT.
https://www.youtube.com/watch?v=IF_BnlirgxE
http://www.visitjapan.jp/en/

But also 歓迎(kangei), いらっしゃい(irasshai) and おかえり(okaeri) mean "welcome" in some cases?

Craig,

What would you want to use it for?

If it's going to be for a banner at a party welcoming someone new, for example exchange students into your school, or new associates into a work place, ようこそ followed by "XXX へ (welcome to XXX) would be usual.

歓迎 can sound/look a bit formal, but can be used instead.
いらっしゃい is more like store staff "welcoming" customers coming into their establishment.
おかえり is "welcome back home," when someone RETURNS to where he belongs, so to say.