Okay if you have no time watch this. It's a 40 minute recap of season 1 that explains all major plot elements. It's narrated by John Barrowman.
https://www.youtube.com/watch?v=N06bJ23bOms
If you have more time than here's a list of season 1 episodes you can watch, but if you're really pushed, just watch the recap
Episode One- Pilot
Episode Three- Lone Gunman
Episode Four- An Innocent Man
Episode Five- Damaged
Episode Six- Legacies
Episode Nine- Year's End
Episode Ten- Burned
Episode Thirteen- Betrayal
Episode Fourteen- The Odyssey
Episode Sixteen- Dead to Rights
Episode Twenty-One- The Undertaking
Episode Twenty-Two- Darkness on the Edge of Town
Episode Twenty-Three- Sacrifice
Now season 2 had a more continuous arch, and I wouldn't really skip any (well except for a few), but if you really have to here are essential episodes I think
Episode One- City of Heroes
Episode Five- League of Assassins
Episode Seven- State v. Queen
Episode Eight- The Scientist Introduces the Flash!
Episode Nine- Three Ghosts **Important
Episode Fourteen-Time of Death (I only included it because it ends in a cliffhanger that leads into The Promise, This episodes kind of sucks lol)
Episode Fifteen- The Promise
Episode Eighteen- Deathstroke
Episode Nineteen- The Man Under the Hood
Episode Twenty- Seeing Red
Episode Twenty-One- City of Blood
Episode Twenty-Two- Streets of Fire
Episode Twenty-Three- Unthinkable
So Total: 13 episodes
Hope this helps!