You should be using abilities from all three of your class tree.
There are 300 different ways to play Sorcerer solo, and many will have their own opinions, here's mine. While pets are not suitable for PvP they can help with survivability in solo PvE.
Daedric Summoning: Unstable Clannfear and Summon Twilight Matriarch
Dark Magic: Crystal Fragments
Storm Calling: Liquid Lightning
Destruction Staff: Force Pulse
Ultimate: Summon Storm Atronach (until you get Shooting Star)
As I mentioned, 300 different ways to play it. I suggested that build due to its high survivability during leveling. Swapping Inner Light instead ofTwilight Matriarch will increase your crit chance, and therefore DPS, and therefore speed to kill. But its at the expense of survivability. If you find you seldom go below a safe health threshold (33% to 50% is my safe) then you can afford to mix up your build a bit.
The build gets very boring if you swap out Liquid Lightning and add Inner Light. All you do is spam force pulse until Crystal Fragment procs. Its effective, but slow and boring.