There are end number of things to do and see in and around Darjeeling and an end number of agents available to take you for the seven points and 5 points sightseeing - that's is the normal jargon for mainstream sightseeing in Darjeeling that will include - Lebong Race Course, Tiger Hill, Tenzing Rock, Himalayan Mountaineering Institute, Zoological Park, Peace Pagoda/Japanese Temple etc.

However, outlook of travellers are changing - rapidly. They want to see and do things that are not exactly popular or mainstream.The suggestion for those seeking something different during their time in Darjeeling would definately start with walks around.

Great walks around the town includes -

 Aloobari Walk: Start the day (fill your stomach with breakfast first) from Chowrasta towards the pony stable, take theway that is in the middle (ask anyone for Aloobari and they will point you towards it), th walk will take you to Tungsung, a crowded inhabited area but offering you glimpse of something that is not touristy at all, a real world of Darjeeling locals, the walk will lead you to Buddhist cemetary and towards Aloobari Gompa (Monastery) also locally known as Yolmowa Mak Dhog Monastery - this monastery was established during 1914, isn't in a good and rather glamorous condition (which normally is the case of monasteries), but that is where the charm lies. It could be closed but if you wait and find out then someone from the village can open the doors for you. The monastery is magnificent inside. Carry on the same same path and it will lead you to Jorebunglow, half km away from Ghoom Railway Station where (if you are lucky and on time) can catch a Toy Train to Darjeeling, or make your way to Ghoom Monastery, the oldest monastery in Darjeeling (locally known as Yiga Choeling Monastery). This walk can easily take half day. 

 prayerflags on the way to Ghoom oil lamps in monsatery

Aloobari - Jorebunglow - Tiger Hill Walk: A full day adventure. Do the Aloobari walk first, then from Jorebunglow take left (main road) till you reach the junction (ask a local for direction to Tiger Hill), the climb will start immediately and you will feel running out of breath, this is normal and your body will adjust. The increase is altitude is noticable. Avoid taking the motor road, instead take the pathways from the left, it is narrow pathways used normally by the localls. The flora and fauna will amaze you and the view of the mountains are incredible from here (on a clear day). When you start to reach the top, the climb will gladually end and a flat walk starts, you have reached a meadow like landscape, this is where the locals normally comes to picnic once a year (on new years day), if you happen to be around the time, make your way here and you will have free food and people to talk to. The walk till Tiger Hill (observatory tower) is another steep climb of about 1.5 - 2 kms. This is going to be hard for some people. Almost half way through, you will see the most revered temple called 'Sinchel Devi Mandir' on your left hand side, slowly clim  towards uphill and you on the highest point of Darjeeling hills (apprx. 2500 mts). On a clear day, you can see the Everest and Makalu from here.