There's always a so-called start dash campaign, which is a bonus to the daily login reward for new accounts. Just make sure to login daily to get it. Besides that:
Get a transfer code. Go to "My Room" (the rightmost icon in the bottom bar) and there should be an option that says "Get Transfer Code" or something similar. You'll have to set a password too. Write down the transfer code somewhere safe and remember the password. This is your save game. It is linked to your account and tracks any progress you make. If your device breaks etc., you'll need to put the transfer code and the password in to recover your account after a new installation. Transfer codes only work a single time, so after a transfer, you'll need to get a new code. If worst comes to worst and you don't have a transfer code, there's another recovery procedure (see the sidebar "NA/JP recovery guide")
Clear Fuyuki. Many events and rewards require you to have the Fuyuki cleared to participate, and it doesn't take too long.
Do the weekly quests. Just tap the clipboard at the bottom to see them.
Every day, you get 10 friend point summons for free, but they expire if unused. They are pretty valuable at the start.
Every month, the Shop restocks. The one thing you want buy out pretty much every month is the Summon Tickets in the Mana Prism shop. (You get Mana Prisms by clearing the daily quests in Chaldea Gate for the first time each day.) Don't stress yourself about it in January, but keep it in mind for the following months