There are 4 entrances to the underdark I can think of (and I feel like I'm missing some):

1 - fall in from the phase spider cave. Hard route, as you land in a pretty tough encounter
2 - Teleport from Auntie Ethel's cave. Although I have yet to be able to get through that door, it seems bugged.
3 - There is an elevator from the Zhent hideout. This gives you much better positioning on the hard encounter you drop into from option 1.
4 - You can stroll in from the basement under the goblin camp/old Selune temple.


1 is the roughest way to go, and that's the way the OP is trying to go.

I would wait until the party is 4th level before tackling the underdark.