Between June and early October you can make an overnight trek to Tarsar Lake from Aru or continue on to nearby Masar and Sonasar alpine lakes. Longer treks go via Lidderwat to Sikwas and Sanmus to Sumbal near Sonamarg, or to the Kolahoi Glacier; for these you need to have a reliable agency. Always check the security situation as militant activity could potentially affect the area.
From Chandawari in the East Lidder valley it's an easy day hike to Sheshnag Lake.
Most guesthouses in Aru can find a horse and horseperson for around ₹1000 per day, or can arrange full treks.