You are correct in saying that the total spin for two electrons is S = 1 as its spin multiplicity is 3 and that L =1. The J value is found in the same way as S and L values were using the Clebsch-Gordon series, $$J= |L+S|...|L-S| $$ This means find the maximum value, then the minimum make it positive, and separate any other values by 1. Sometimes there is only 1 value so be careful. We use this type of series as a short-cut for vector addition of angular momentum which is why you question '1+1 = 1'.
In your case J = 1+1 to 0 , which gives three values J = 2, 1 and 0 so three possible term symbols. There will also be singlet terms where the total spin S = 0.
The triplet term means that electrons are in different orbitals, generally this lowers the energy, and also means that there is no Pauli principle to consider. Hund's rules are used to determine which levels and states are lowest in energy. (See for example Steinfeld, 'Molecules & radiation')