THUNDER CROSS II / Konami / Arcade
More of the same in this sequel, but since the original offered very solid shooting action in the Gradius mold, that's totally fine. This one ups the ante a bit with even prettier and more elaborate backgrounds plus a little higher level of challenge than the prequel did. Both are worth playing, but this one stands out a bit more.
Some new stuff to look for here is a couple levels with branching paths, and bosses in general being a little more resilient than before (though they still tend to fold like a cheap tent fairly quickly if you can keep powered-up fire on them). Also, while this game continues in allowing you to continue right from where you lost your last life with a quarter, be aware there's an unadvertised continue limit (10 or 12 or so) which might be a VERY rude surprise when you get to the later levels.
Not as easy as the first game, so it's less of a good introductory choice for beginners you want to co-op with, but it's not too over the top and still a good choice for those who enjoy shooters but aren't super hardcore about it. And the soundtrack is excellent