The Tigre Delta is one of the most popular day trips in Buenos Aires. You can easily go on your own using public transportation. Take the train to Tigre from downtown Buenos Aires and from there, take a boat into the delta. There are public transport boats (lanchas), private water taxis, and big catamaran tours.

My recommendation is book a private tour. The huge catamarans only scratch the surface, giving a very brief history via loudspeaker.

Going on a smaller, private boat will let you go onto quiet canals and to empty beaches with lunch included. It'll be a trip highlight!
But if you're on a tight budget, go on your own. Take the train to Tigre from Retiro and public transport into the delta via lanchas colectivas. You can take a boat out to Isla Santa Rosa for a relaxing walk among stilted homes or out to El Gato Blanco, a parrilla in the delta, for lunch.