1. To my knowledge, there is only one Harmonic Series, and it diverges (slowly). Therefore it has no sum.
2. There are many types or classes of infinite series: geometric series, telescoping series, power series, Taylor series, E-Series, series of functions, Laurent series, and on and on. There are some guidelines for summing each kind of series, (there are even strategies for "summing" certain divergent series), but there is nothing like a general formula that simply spits out a sum for whatever series you input. One learns different classes/types of series, and some general strategies for solving them. In some cases, quite a bit of ingenuity is required to demonstrate that a given series sums to a given value.
If you have a specific series in mind that you need to sum, please provide it and one of us may be able to point you in the needed direction.