How do I find the oxidation state of each element in a compound when I'm only given Oxygen or Hydrogen?

In NO2- N has o.n. +3
In fact in HNO2 N has o.n.
+1 + x + 2 (-2) = 0
x = +3
In Ni(NO2)2
we have
x +( 2 x 3) + ( 4 x -2) = 0
x + 6 - 8 = 0
x = o.n Ni = +2

This is an ionic compound with a cation and a polyatomic anion.  You know the total charge on the NO2 is -1 so do this one individually.  
2(-2)+x=-1
-4+x=-1
x=3

The charge on the Ni atom is equal to its ionic charge: +3
So in conclusion:
N=+3
O=-2 (always)
Ni=+3

Firstly note that for a complex ion (e.g NO3-, NO2-), the sum
of the oxidation states of the elements will equal the overall charge of the ion.

So for Ni(NO2)3, you must either remember or look up the ionic charge for the nitrite ion (NO2^- ).

So since the overall state of the nitrite ion is -1, the oxidation state of N must be: x + -4 = -1.  

So x = +3.  You only need to consider one nitrite ion as they are all identical.

You can also find Ni oxidation state: A nitrite ion has a -1 charge, there are 3 nitrites attached to one nickel, so to make it neutral overall Ni must have: x + (3 x -1) = 0.

Hence the oxidation state for Ni is +3.

Common ions to remember include sulphates (SO4^2-), carbonates (CO3^2-) and nitrates (NO3^-).