Your approach of finding infinite quotients is certainly a standard one. There is, however, a slight tweeking of it that helps in the event that this approach breaks down - search through some low index subgroups. If any of these have infinite homomorphic images then your group must posses an infinite subgroup and thus must also be infinite. In Magma the command "LowIndexSubgroups" can do this and I suspect somthing similar works in GAP, matlab etc.
As with the other techniques this is not a sure-fire 100% guaranteed method, but it is sometimes useful.
Simplicity of an infinite group is a much harder question to address. Needless to say that if I was a beting man then I would certainly put money on your group not being simple.
