It's a new day today.

Today is the beginning of a week.the sun is shine.good lucky for everone

Today is the beginning of the week. The sun is shining. Good luck to everyone!

In this context, it would be "the week" not "a week". It would be be awkward otherwise. Shining is correct here, because to answer what the sun is doing, it is "shining" the -ing is adding the action to the sentence. Good luck to everyone is directing the luck; for everyone works, but I feel like "to" works better.

I hope this helps =D!