Spituk falls before Leh when you are coming to Leh through the Srinagar-Leh highway. Spituk is just 8km away from Leh. Take the left road from Spituk straight. On your right you will see a gate mentioning "Matho". Take that road to reach Matho.
If you're just starting out with trekking, start with this relatively easy and short (4 days) trek, which covers some scenic areas of Leh. Start by driving from Leh to Spituk village, which is about 7 kilometres away. The hike from Spituk to Zinchen (usually the first halt) could take around 4-5 hours, going through dry, barren land to greener areas of Zingchen valley. Here the Rumbak River flows into the Indus and it's the perfect place to set up camp, enjoy the landscape, and maybe even take a dip in the river. The next day will take you through Hemis National Park where you can spot plenty of mountain flora and fauna. A 3-4 hour trek, largely uphill, will bring you to the pretty village of Rumbak. Day 3 will be a bit more arduous with a steep trail near Stok La and then descending down towards Mankarmo (5-6 hours of trekking). On the last day a 3-4 hour descent through narrow valleys with lovely rock patterns will bring you to the village of Stok. Before heading back to Leh be sure to check out the 19th century Stok Palace, which now houses a museum.
2. Markha Valley