Hey! I just joined MyFootballNow recently. When the new season opens, can I trade? When does trading begin? And can I do trades of draft picks on draft days?

Since u are a new player u have to wait until ur account is one month old to trade. It prevents new players from making I'll advised trades or giving away too many high quality picks for small returns.

And after the first month, can owners trade away draft picks on draft day?

You can trade at any point except from Reg Season Week 8 - End of Season .. after your 30 days of course.