First off, her important normals.
s.LP is great. Its her fastest normal but it won't hit on Filia or Valentine while' their crouching so becareful of that.
cr.LK is probably her best combo starter because its a low and pretty fast.
cr.MK is her longest range normal on the ground.
cr.HP is great because the flame isn't part of Squigly's body and can't be hit so it stuffs all kinds of moves. It also hits low and has pretty long range. You can combo into Silver Cord for a full combo but you have to wait until all the hits on cr.HP connect.
s.HK is your launcher. You use it to start air combos.
F+HP is Squigly's overhead. If your opponent is blocking low a lot, its super useful. Cancel the last hit into Silver Cord for a full combo afterwards.
j.LK is very fast and has good range for a light attack. Its one of her best air normals
j.MK hits all around squigly's body. It has multiple hits so its easy to start a combo from. Good for air-to-air and jump-in attacks.
j.HP is great for keeping your opponent in check in the air but it shouldn't be over-used since Leviathan's head is vulnerable during the attack so its easy to beat if your opponent sees it coming.
Important Special Moves
Arpeggio QCF+LK - If you happen to do a normal attack that's unsafe on block like s.HK, you can cancel into this move to stay safe and maintain pressure. Its great for when you're not confident with your stance cancels yet.
Silver Cord QCF+MK - Its a great tool for starting a full combo from a stray hit.
Tremolo QCF+HK - Its probably her best combo ender in most cases since it ends in a knockdown and does good damage.
Fallen Woman QCF+K (air) - LK and MK versions of this move are great for air-to-ground pressure and the HK version can be used to start a full combo from an air-to-air hit but be careful. The HK version is very unsafe if blocked.
Super Moves
Daisy Pusher QCB+LP+LK Its a very easy to avoid throw but it has lots of invincibility. Use sparingly.
Squigly Battle Opera QCB+2K - A very powerful super move that is useful for just about everything. It puts a projectile on the screen that stays in a fixed position relative to the camera. You can use it to start combos, to apply pressure to your opponent or to keep them away from you.
Inferno of Leviathan QCB+2P - Costs 3 meters. Shoots a tracking fireball that does less damage the longer it stays on the screen without hitting something. It can be used as a very powerful anti-air attack that can lead into a full combo, or you can throw it onto the screen and force your opponent to avoid it while you move in.
Typically on the ground you'll be using cr.LK > cr.MK a lot to poke to try to get a hit that you can start a combo from. If you're pressuring your opponent up-close use F+HP to keep them guessing about where to block. Her throw is also very good since it has pretty long range, if you're not close enough to the corner for her throw to wall-bounce then cancel it into QCF+LK+MK to juggle them after the throw so you can follow up. When your opponent blocks your attacks, cancel into QCF+LK and QCB+LK+MK to stay safe and maintain pressure.
In the air you'll want to be doing j.LK > j.MK and when it hits you can do QCF+HK to start a combo. From longer ranges, j.HP is useful but sometimes risky. If you're having trouble getting close to your opponent, you can use j.HP to try to force them to come to you instead.
or 
. If your opponent blocks, you can only use a stancel if you do the full motion, you can only shortcut it on hit. 
