Some good methods to look into:
Fridrich is the most popular method, it requires 78 algorithms and not too much thinking and is very fast.
Petrus is a block-building method meaning that it has a low move-count compared to Fridrich but is harder to get very fast with.
Roux is a different method that some people here use, it involves both block-building and a good number of algorithms (not sure exactly but I think it's a bit less than Fridrich)
ZZ is a less popular method that is designed to use very fast moves, I don't know as much about it but you could look into that as well.
Those are the most popular methods, some others are Heise, Waterman, Ortega and Guimond. The four I explaiend about would probably be your best bets to do speedsolving with easily. My personal preference is Fridrich.