I will keep it concise but informative-
How to reach: No matter from where you start, you will have to reach Silliguri (in West Bengal) first. Prefer train journey. The railways station of Silliguri is known as 'New Jalpai Gudi'. Alternatively, one can take a flight to Bagdogra - the nearest airport from Gangtok. From Silliguri, it is 4 to 5 hours journey to Gangtok by bus.
What to See: A good number of ancient Monasteries (Rumtek, Enchey and others) and Institute of Tibetology. Do not miss Banjhakri Falls which is more of a water park. Take pain to get up early to reach Tashi View Point and enjoy the fantastic view of sun rising above the Kanchenzonga hills. Ganesha Tok and Hanuman Tok (temples basically) are really peaceful. You will get a nice view of the whole Gangtok from Ganesha Tok. There is one Botanic Garden which is good to see if you pass by it, nothing very special. Explore Institute of Handicraft where you will find a variety of things to take back as souvenir. It would be wise if you compare prices here with those in local market (which you will anyways see while roaming on MG Marg) as you can come back here again since this place is at walking distance from the center of Gangtok. By the evening, be all geared up to roam around MG Marg which is one busy (and happening) place in the town. Good restaurants, major shops, hotels, Sikkim tourism office, ATMs and the overall atmosphere-All these things make this place a perfect hangout venue in the evening. Extend your walk on MG Marg to adjacent local markets like Sadar Bazar and Lal Bazar, and you may find cheaper deals here (if you are good at negotiating with small shop owners :-) ).
Closing Remarks: A calm and beautiful destination. Full of refreshing natural scenery, both on way and around it. People are friendly, helping and soft spoken. It takes a complete day to explore the main attractions near Gangtok. If you have more time, you must extend your trip to Nathula Bypass, Tsomo Lake and Old Baba Mandir (towards east border of Sikkim) or Namchi and other nearby places (towards south border of Sikkim). My overall rating of Gangtok is 4.5/5.