I am building an edh deck for about 5 dollars and was wondering, If I cast Sickening Dreams dealing more than enough damage to kill my opponent and I at the same time, who wins?
It's a draw. Whenever all remaining players lose at the same time, the game ends in a draw.
From the Basic Rulebook, under "Winning the Game:
If both players would lose the game at the same time, the game is a draw—nobody wins.
Similar information is repeated under the glossary for "Life total" and "draw the game".
The game is a draw. MTG's rules are not designed to try to prevent draws. This is perhaps most evidenced by the card Divine Intervention which can specifically cause the game to end in a draw.