Your choices are:
Jamestown
An introduction to bullet hell staples like rapid vs. focus attacks, interesting scoring systems, and of course, bullets a-plenty. Your progress is saved between levels, and there are multiple difficulty settings, allowing you to get your feet wet before really taking on challenge. Jamestown features great stage design, a welcoming difficulty curve, and multiple playable ships.
Metal Black
A 1991 arcade/Saturn horizontal shooter originally planned as Darius III. The game revolves around a simple system where mini DNA power-ups constantly swarming the screen slowly fill a power meter. Meter level increases the power and size of standard bullets and determines the form of the player's bomb-like hyper laser, which itself empties the meter and resets power to 0. The coolest part is that bosses power up / use hyper lasers / power down the same way you do, so DNA collection is a constant tug-of-war. The art and music are also fantastic.
Sine Mora
The theme for this game is time. You have a limited amount of time to get to the end of the stage. Once it runs out, the game's over. "But its an auto-scroller! How am I supposed to race the clock?" you ask. It's simple! Beating enemies increases your time limit and there are special abilities that can manipulate time in your favor. Your ship wont go kaput until the countdown ends so you dont have to worry about death-on-impact but every hit you take will make the countdown go faster.
Zero Wing
The meme game, you all know this one.