Kashmir is usually sold as a single destination, but it's really three towns strung together, each with a different character, and the trip works best when each one gets enough time rather than being reduced to a single rushed day.
Srinagar is the base — a sunrise or sunset shikara ride on Dal Lake, a night or two on a houseboat, and the Mughal gardens at Nishat Bagh and Shalimar Bagh, terraced lawns and chinar trees running down to the water. The old city, around Jamia Masjid and the Zaina Kadal bridge area, is worth a slow walk if you'd rather see the working city than just the lakefront.
Gulmarg is built around the gondola — phase one runs up to Kongdoori, phase two continues to Apharwat Peak at around 13,000 feet, and on a clear day the views run out toward Nanga Parbat. It's a ski destination between December and February; in summer, the meadow around the town turns into one of the highest golf courses in the world.
Pahalgam trades the crowds for valleys — Betaab Valley and Aru Valley are both short drives from town, good for a half-day each, and Chandanwari, the traditional starting point of the Amarnath Yatra, makes a scenic final stop even if the yatra itself isn't the plan. The Lidder river runs through most of it, cold and fast enough that trout fishing is a genuine local activity, not just something offered to tourists.
On timing: April to June brings the greenery and mild days most people picture when they think of Kashmir; December to February is for snow and Gulmarg's ski season, with Srinagar noticeably colder and quieter. Six to seven nights is a realistic minimum to give each town its due, and since the towns themselves are close together — Srinagar to Gulmarg is about 50km, Srinagar to Pahalgam about 90km — the road days are short. Book the Gulmarg gondola tickets ahead of time in peak season; the queue at the base station can run into hours if you turn up without one.