If you can only make one stop on the "Hemingway trail" while in Havana, this is the one to make! This is his home in the country outside Havana (about a 30-45 minute cab ride - ask for Finca Vigia or Museo Hemingway), and it's a really tranquil, interesting way to spend an afternoon! Be sure to go on a nice day though. In order to preserve it, you can't actually go inside his house, though you can look through the open doors and windows on a clear day, climb up to his tower, check out his pool and his fishing boat, the Pilar. This was one of my favorite museums (and favorite spots) in Cuba!
Hemingway is supposed to have said "My mojito in La Bodeguita del Medio and my daiquiri in the Floridita", so I recommend you have a daiquiri at the Floridita when you visit! (I'll discuss la Bodeguita in a minute). The restaurant is large and very nice inside, the bartenders are skillful, the drinks are strong, and there's even a bronze sculpture of Hemingway leaning on the bar in the corner! This place also calls itself "the Cradle of the Daiquiri", though I don't think it was invented there, but elsewhere in Cuba. While this is a fun stop, I recommend you just have a drink and listen to a little music, because the food is pretty pricey and the place will be full of other tourists!
Hotel Ambos Mundos is a nice hotel in which Hemingway stayed before he moved to his home at Finca Vigia. He stayed in room 511, which has gorgeous views of the Plaza de Armas. Calle Obispo is the street on which this hotel is located, and it's one of the best shopping streets in Old Havana to this day. Be sure to take a stroll down it, take a peek in the sherbet-colored Hotel Ambos Mundos, and maybe check out the little book and antique market that's usually going on in nearby Plaza de Armas!
As mentioned above, Ernest Hemingway is said to have preferred the mojitos at La Bodeguita del Medio. This is just a small bar off a side street from the Plaza de la Catedral, but it's a big stop on the Hemingway-interested tourist trail! The place was a bit small and crowded for more taste, but they had signatures of visitors from all over the world on its walls and it's a nice place to cool off with a mojito while you're out exploring the plazas of Old Havana!
The final place on my list, which, unfortunately, I was not able to visit myself, is called Cojimar. It's the small fishing village to the east of Havana where Hemingway used to go fishing, and which served as the backdrop and inspiration for The Old Man and the Sea. Next time I go to Cuba, I'm definitely heading out there to check out the village, the cool little castillo along the coast, and eat at Las Terrazas restaurant there. From what I've read, it's about a 45 minute cab ride from Old Havana, but I think it could make a great stopover on your way to or from the fabulous Playas del Este. Above is a painting of the Cojimar fortress that was on display at Finca Vigia when we visited. Cojimar definitely tops my list of places I need to go next time I visit Cuba (which will hopefully be soon!).