Playa El Doradillo is famous for whale watching from its shore. There is a major drop off just off shore and this deep water allows the whales to get as close as 10-20 meters from the shoreline.
Access to Playa El Doradillo is free so this is an excellent option for travelers on a budget.
As you drive along Ruta 42 you'll see a lot of "bajadas" or beach access points. Playa La Cantera is the most recommended for whale watching, here is the pin for it on Google Maps.
What time of day is best? Plan your visit for high tide so that the whales will be able to get as close as possible to the beach. The best times are generally in the morning and late afternoons when the water is stiller and the tide is high but if you want to reference the tide table you can do that here.
How to get here to Playa Doradillo? By car. There is no public bus so you'll either need to rent a car, which I recommend for this region. If you don't want to drive, you can hire a taxi/remis from downtown Puerto Madryn. It is about a 20 minute drive from the city.