If you take a flight in, fly from Delhi (DEL) Airport to Dehra Dun (DED) Airport. This journey will cost USD 22-95 and it will take about 50 minutes. From DED, you can take a private hire to Rishikesh. This journey will take 17 minutes and cost you about USD7-9. Private cabs will be needed for further transportation.
Alternatively, if you come by train, then you could train from Indira Gandhi International Airport (IGI) to Haridwar. You would have to board the subway Orange Line from IGI to New Delhi station. This would take 30 minutes and cost USD1. Then make a transfer at New Delhi station and board the train towards Ghaziabad. This journey will take you 35 minutes and cost you USD1-12. Please check the daily schedules beforehand to know which train you should be boarding at what time. From Ghaziabad, the train costs USD4-15 and takes you 10 hours 19 minutes to get to Rishikesh train station. Do note that the train from Ghaziabad only runs once daily. Pick-up is arranged from Rishesh/Haridwar for the respective treks.