Mulligans
General Mulligans
You don't need to keep The Storm Bringer, but you do want to keep Electra, especially with the 3 cost spells. . Keep Elementary Reaction only if you have an Elemental in your hand, and Stormforged Axe if your playing against aggro, so generally not druid or priest
Against Aggro: If you don't have any 1 or 2 cost minions, or the weapon, mulligan away Electra. Electra is too slow to hold and you can get overwhelmed by aggro pretty quickly. Try to hold tightly to Storm Chaser, as it gives you your only win conditions.
The Boomsday Project brought about an interesting new card, The Storm Bringer. Utilising the power of Thunderhead, Primalfin Totem and Primal Talismans flood the board with as many minions as you can, then Thrall, Deathseer or The Storm Bringer can come in. Far Sight Is probably the most powerful card Shaman has ever got, it is able to reduce the top card of your deck by three! A two cost Bloodlust or four cost The Storm Bringer will be a game changer for sure.
Storm Chaser Can get you Bloodlust and The Storm Bringer, so its a good card to keep in your opening hand, if you have early game minions like Fire Fly Mecharoo Menacing Nimbus and Primalfin Totem.
Electra Stormsurge is for Far Sight Primal Talismans Bloodlust and even Lightning Storm if your struggling a bit. Casting The Storm Bringer twice is useless so try to avoid doing that.