The Time Dilation feature is your best friend. While active, it slows down time allowing you to dodge / shoot enemies. In this state, you can move slightly faster than your surroundings to help give you a better chance to survive.
Each ship has its advantages and disadvantages in regards to weapons. When upgrading in the store, be sure to upgrade the skills that correspond to your ships best weapons.
Power-ups are the first key to victory. When you kill an enemy and it drops a power-up, don't feel rushed to pick it up right away. Sometimes, killing enemies or waiting for them to pass will help give you a better chance to stay alive while getting the power-up.
There are four levels for each power-up type, so when an enemy drops one, be sure to wait for it to cycle through until you find the type you want to upgrade. Simply flying over to it as soon as it drops may not give you the helping hand you were looking for. If your missiles are already at the max power-up level, there's no sense in getting another missile power-up.
Level memorization is the second key to victory. The game doesn't require you to memorize enemy formations and level layouts, but it can certainly help in the later levels. So if you feel stuck, use the tips mentioned above and try your best to remember where the enemies are going to come from next.
Drones for the win! Having trouble staying alive with only three lives? Well, purchasing drones in the store could keep you alive long enough to move on to the next level. Not only do drones take a bullet for you, in essence, giving you an extra life, but they can also fire the weapon type you're currently firing.
One nuke to destroy them all. Although you can only carry three at one time, they'll certainly help you out in those moments where you just want to close your eyes and hope you make it out alive. Nukes have the ability to destroy all bullets and enemies on screen at the instant they hit the ground.
Good luck and don't give up!