There are various possible route. The two most comfortable are:1) Narita Airport - (bus) - Utsunomiya - (JR Nikko Line) - Nikko - (bus) - Lake Chuzenji
Approx. 5 hours and 6000 yen one way
2) Narita Airport - (JR Narita Express) - Tokyo - (JR Tohoku Shinkansen) - Utsunomiya - (JR Nikko Line) - Nikko - (bus) - Lake Chuzenji
Approx. 4.5 hours and 9500 yen one way
Route 2) is covered by the Japan Rail Pass, except the bus to Lake Chuzenji. But a Japan Rail Pass does not pay off for just a side trip to Nikko. You would have to make other, long distance trips, as well.
There are also considerably cheaper routes, e.g. by using Tobu Railways. The following page can give you an idea:
http://www.japan-guide.com/e/e3805.html
Do they allow luggages?
Reasonable amounts of luggage is allowed on all the above mentioned trains and buses.