Costa Rica has two main seasons: dry (verano, December through April) and wet (invierno, May through November). "Best time to visit" depends entirely on what you're after.
Dry season (December–April) is peak tourist season for the Pacific coast, Guanacaste, and the Central Valley. Roads are passable, beaches are sunny, and hiking trails are accessible. It's also when prices are highest and popular spots like Manuel Antonio and Arenal are busiest. Book accommodation well in advance for December, January, and around Easter (Semana Santa), when Costa Ricans travel domestically in large numbers.
Wet season (May–November) brings afternoon rains to most of the country, but mornings are often clear and the landscape turns lush and green. Prices drop, crowds thin, and wildlife viewing is excellent. Rivers run high, which is ideal for white-water rafting (the Pacuare and Reventazón rivers are at their best). July is often a dry pocket on the Pacific slope. The Caribbean coast follows its own rhythm: heaviest rains in November–January, drier in February–April and September–October.
Turtle nesting peaks at different times by species and location. Leatherbacks nest at Tortuguero and Playa Grande from March to July; green turtles at Tortuguero from July to October; olive ridleys at Ostional (Nicoya) arrive in mass arrivals called arribadas, most reliably July through December.
October is worth singling out: it's Carnival month in Limón, one of the country's best street festivals. The rainy season is also well underway, but if you're heading to the Caribbean, it's a fine time to go.
The Central Valley — where San José and most cities are — has consistently mild weather year-round, usually in the low 20s Celsius (low 70s Fahrenheit), so it's never really a bad time to be there.