I fish a lot on the island I didn't see the tan until the next day of playing. just play on the island fishing and bugging also I play a lot of the island games I don't know if you can get a tan in the mini games but you never know and do I a lot of diving maybe you tan while swimming because I have done a lot of that too I wish I knew how I did it.
The tanning starts out very subtly. I didn't notice I was tanning until I went to my friends town who didn't have the island unlocked yet. We were hanging out and I noticed how pale he was compared to me! That was a few days ago and now I'm even darker than before! Might have to bust out the umbrella on sunny days to keep from tanning too much