For trail/running in cold weather, you'll need rather thin, breathable gloves. Depending on your degree of sensitivity to cold, you can use the same gloves for ski touring, hiking and trail running, as these gloves are quite versatile.
For climbing and cycling, you'll need gloves with tougher, abrasion-resistant materials, or harder shells to avoid impact on your hands. You'll also find a choice of mittens for your climbing gloves, to keep your fingers sensitive to the holds.