Assumed you come from Luxembourg city, you can get there by:
Bus: The bus lines 111 and 110 take you from the central station and the main bus station (Aldringer) to Echternach in under one hour.
Car: Leaving the city via Kirchberg, the directions to Echternach are well explained.
Bike: Via Kirchberg, you got a perfectly signed cycling-trail which should take about 3 hours.
You can also make a trip here from Vianden, it takes about 40 minutes - get the 570 bus to Ettelbruck (get off when you see the "Gulf" petrol station), then cross the road and take the bus to Echernach. The connection time is generally short (about 10 mins), but best to get your hotel/hostel to check times before you go.
The old abbey founded in 698 by St. Willibrord is one of the must-sees in Echternach. Today, it's a school building for high-school classes (7th-13th grade). The basilica next to it still serves as church of Echternach. If you're interested in the history of the monastery, there is a small but nice museum next to it.
The town square with its square cross and the historical guildhall "Denzelt" is a nice place to see and stay.
The roman villa about 20 minutes walk from the centre shows the remains of an ancient roman house. A walk-through is free, worth a visit.
The lake is an artificial lake. Enjoy the silence around it, 20 minutes walk from the city centre.
Youth Hostel Echternach, Chemin vers Rodenhof, L-6487 Echternach, ☎ (+352) 26 27 66 400, . checkin: 17:00; checkout: 10:00. 24,20 Euro per bed in a dorm, breakfast included. All rooms are ensuites. The hostel is very focussed on sports activities and has its own 14 metre high climbing wall, bikes for rent, trampoline park, etc.edit