If you don’t have a JR Pass, the cheapest way is to take the TOBU Railway train at the Tobu Asakusa station.  To get to Tobu Asakusa station from Shinjuku, take the Yamanote Line to Ueno Station, then transfer to an Asakusa-bound train on the Ginza Subway Line. The last stop is Asakusa Station.  Tobu Asakusa Station is located in a surface station that is connected to a department store. The best way is to just surface and walk, there are plenty of signs directing one there and shouldn't be very difficult to find.

This is a very long day trip, taking you at least 3 hours each way for both the metro ride from Shinjuku to Asakusa (45 minutes) and the TOBU train ride (1:45 minute with Spacia train)with all the transfers.

Tobu railway offers passes which includes return (rapid and section rapid) train tickets and admission to the sites.  To take the Spacia express train which is a more luxury train and goes directly to Nikko you have to upgrade and pay about 1000 yen more.  The Spacia train is much nicer and faster, totally worth the money.  

Spacia trains (limited express) are named Kegon or Kinu.  The only direct train to Nikko is the "Kegon." If you want to take the "Kinu," you have to change trains at Shimo-imaichi.  The train to Nikko leaves the same platform of the opposite side at Shimoimaichi Station (Travel time from Shimoimaichi Station to Nikko Station is about 8 minutes.)

The Tobu railway’s World Heritage Pass (3600 yen) does not cover the bus ride to Lake Chuzenji.  The All Nikko Pass (4400 yen) includes the bus ride to Lake Chuzenji but does not include admission to the temples and shrines.  If you get the All Nikko Pass, buy the combination ticket (社寺共通拝観券, ¥1,000) that covers Toshogu, Rinnoji, and Futarasan.  You can reserve the Nikko passes on line and pick them up at the Tobu Asakusa station tourist office.

At the Tobu Nikko Station in Nikko, visit the tourist office to get basic maps and information. Then you just hop on the World Heritage Bus (fare included in either pass) that goes around Central Nikko, and the bus stand is right in front of the Tobu Nikko Stn.  It takes about 10 minutes bus ride to get to the site where the temples are. Once you get inside the complex, it's entirely walkable. Allow about 3 to 4 hours for the site.

After your visit, you can head back out to the main road. Go across and find the bus stop (or ask somebody in a shop where the bus stop is) in the direction of Lake Chuzenji and Kegon Falls.  It takes about 40 minutes from the temple complex to the lake.  The lake itself is beautiful and the waterfall quite impressive. There isn't anything much at Akechidaira Ropeway, so you may want to skip it. Allow two hours, or more should you want to walk around the lake some more or visit more shrines. At the end of the trip, just take the bus back down (take a motion sickness pill if you are prone to this problem since the road is filled with hairpin turns) all the way to the station for your trip back to Tokyo.