Quick answer
también = too, also; agrees with a positive statement.
tampoco = neither, not either; agrees with a negative statement.
Tampoco is already negative. Placed before the verb it needs no no at all.
Side by side
Memory trick
Look at the sentence you are reacting to. Positive gets también, negative gets tampoco — mirror the sign, do not translate too/either.
Typical mistakes
—No me gusta. —A mí también.
—No me gusta. —A mí tampoco.
The original statement is negative, so your agreement has to be negative too. También here says the opposite of what you mean.
No tampoco quiero ir.
Tampoco quiero ir.
Tampoco already carries the negative. Adding no in front of it doubles up in a way Spanish does not allow. The double negative only works the other way round: <i>no quiero ir tampoco</i>.
Check yourself
—No he visto esa serie. —Yo ___.