It's always tanning season on the island. It doesn't start in the mainland until late July, I believe.
I like keeping my character pale, so I'm not sure about the tanning mechanics and how to get a dark tan. You do have to keep at it, though. Do it for an hour or so one day, then do it for another hour or so the next day. The tan doesn't start kicking in until the following day, so you won't see results after just those few hours either. Did you have it in Sleep mode? I'm not sure if having it in Sleep mode affects it or not.
To tan, you only have to spend 15 minutes on the island between 10am and 5pm.
You do have to keep your 3DS open, but you don't have to be moving around. You can plug it into the charger and just leave it open if you want!
Make sure you're not wearing anything that prevents tanning!
And you'll only tan one stage each day, so it's barely noticeable until you tan for several days in a row.
Yes, you need to tan during the day. I tan everyday for about an hour on the island. Usually around 12-1pm. You DO NOT see any results until the new day begins at 6am, and you can only have one level of tan each day.