Where I live, this spring has been exceptionally cold and I’m nowhere near acclimatizing any seedlings to outdoor conditions, but I mustn’t forget that many other temperate climate gardeners are enjoying warmer days. If so, and if lugging trays of plants indoors and out is not something that bothers you, it can be advantageous to start hardening off your seedlings, even when nights are still cold.
Days over 55 °F (12 °C) are warm enough for most seedlings you started indoors (annuals, vegetables, perennials, etc.) to begin enjoying outdoor conditions. I’d wait, though, for 65 °F (18 °C) days before putting the tenderest seedlings, like tomatoes, peppers, eggplants, cucumbers, melons, begonias and impatiens, outdoors. It’s highly likely that night temperatures will still be too cold for them to be outdoors, but you might not have to carry them far: you can often simply put them in a garage or shed overnight: temperatures there are bound to be warmer.
This gradual exposure to cool air and sun (called hardening off or acclimatization) will give shorter, denser, tougher plants and may possibly boost productivity later in the season.
Start by acclimatizing the seedlings in a shaded spot that only a few sunny rays can reach, then, after 2 or 3 days of shade, move them to partial shade. Yet another 2 or 3 days later, they’ll be ready for their first taste of full sun.
If day temperatures begin to dip below 55 °F/12 °C (65 °F/18 °C for tender seedlings), temporarily end your experiment with hardening off and move the young plants back to their indoor location. But even a short period outdoors will have benefited the seedlings. You can start acclimatizing a few days later, when temperatures warm up again.
Obviously, don’t plant your seedlings outdoors permanently until there is no longer anyrisk of frost and the soil has warmed up. That usually follows the first warm days of spring by several weeks.