I would recommend Satazius to both fans of the genre and curious newcomers.
Satazius takes the best elements of older games like Gradius and other horizontal shoot-em-ups and improves on them, whilst at the same time jettisoning the bad.
Let's take the weapon system. In Gradius the player could choose their ship build at the start of a credit, while in Satazius the player also gets that opportunity at the start of each level, allowing them to select the best tool for the job at the right moment. Weapons are powered up in a similar way to Gradius, though here each weapon remains powered up as you switch between them. This gives the player the option of focusing on just a select few favourites, or else improving a wider range of weapons at a slower rate. Spreading the love, as it were. Upon death, the player is not stripped of all power like in those unforgiving arcade shmups of yore; instead their weapons are just downgraded a few notches. This means death is not a double punishment, and failing once does not mean you're twice as likely to fail on the next try.
The levels are full of exciting set pieces, high speed chases and environmental hazards. Indeed, the environment is as much your enemy here as the enemies themselves, reminding me fondly of another classic horizontal shmup, Thunderforce III.
Which is a good point to turn to Satazius' few bad points. In those high speed sections the player simply cannot survive if their ship is not a fast one, whereas in tight quarters a nimble ship is easy to crash. The player should be able to switch at will, a la Thunderforce III, rather than having indirect control through power-ups.
And secondly, the music doesn't match the catchy melodies of the above mentioned games, or so many other old-school shmups. But it suffices. That would have just been the cherry on the top of an already inviting cake.