Best time to visit Amritsar is in the winter, between October and March.

By Plane
Sri Guru Ram Das Jee International Airport is about 11km and a 15-20 minutes drive from the city centre. It's one of the modern airports in India and quite adequate if not exactly exciting. Most flights are to Delhi, an hour away.
According to the airport website, both city buses and car rentals are available.
Even the prepaid taxi can be bargained down with the drivers to Rs.300 from an initially ridiculous price of Rs.500 a person (Sep 2016) before you pay, but there is a significant dearth of official taxis or even auto-rickshaws, so prepare for a hard time.
If you would prefer an auto-rickshaw instead of expensive taxi, go left after leaving the airport building. There is a roundabout behind the parking lot where they are standing.

By Train
Amritsar is an important railway station and is well connected to major cities in India through daily trains. Trains can be booked online, at the train station or, most conveniently, at the small booking office in the Golden Temple Complex (located next to Guru Ram Das Sarai).
Here are some useful trains to get to Amritsar:
 

By Car
It takes around 6-7 hours from New Delhi via NH-1. Amritsar is well-connected by bus to most major cities and the northern areas within a days drive. Pathankot is about 2.5 hours away and about 100km away, Jalandhar is about 80km from here, Kapurthala (royal city) is about 65km from here and there are daily direct buses to New Delhi, Jammu, Katra, Chandigarh, Dharamsala (once daily, c. 6 hours), etc.You can find Volvo buses from Chandigarh , Delhi and Katra to Amritsar.