I have fished the beaches in Playa, the beaches and lagoons of Cozumel, and the flats of Sian Ka'an, south to Xcalak.
You can fish the beaches in Playa, no license needed. (A license is "required" if you are fishing from a boat, but in all my years, I've never seen one, and the "price" is included.)
That said, you'd better get up early and watch for morning joggers on your back cast - the beaches get crowded early, and you aren't supposed to be fishing within 200' of people/swimmers - or consider walking a good ways away from the hotel crowds. Those are basically the rules, so don't let any hotel beach guards run you off, telling you otherwise.
I've never caught anything other than small morjarra, (horse eye) jacks, and barracuda from the shore in Playa, but not sure that's really saying much, because you never know. Look for structure of any kind. Fish'll even hang out under the parked boats, just don't hook the anchor rope! See if you can strike up a bargain for quick charter from one of the small snorkel boat captains to run out closer to the reefy areas. Actually, looking at google maps, looks like they have a bunch of submerged blocks for wave breaks in front of your hotel that might hold good fish - grab a kayak!
Sian Ka'an (Boca Paila / Punta Allen) is amazing, and should be on your bucket list, but the road is bumpy-as-all-get-out on a good day. Last time I went two years ago it was particularly pothole-y. Keeps the faint of heart away. I'm getting too old to just run down there for a day trip. Make the drive worth it, stay for a few days, slow down and let the quiet beach life soak deeper into you.
It's easier to take a ferry across to Cozumel and fish the northern lagoon, or jet up north of Cancun to Isla Blanca.