Honestly OP, I would suggest buying Ys I and II on Steam. No offense to the PCE purists.
Far better sprite graphics, awesome PC-88 FM OR REMIXED TUNES, and diagonal movement. Not to mention much cheaper.
I would first play Ys I&II for the PC Engine (sorry, lukester. No offense). You've bought a PC-Engine, one of the most unique and innovative consoles of the 8 and 16 bit eras. Enjoy it. Ys I&II was the reason I pursued this console when everyone I knew was out buying Segas. Understanding the various types of games that were out at that time, Ys is considered to be quite impressive - from it's cinematic visuals to the powerful musical score. Personally, the almost deafening intro made my brain implode, made it difficult for me to enjoy much of the chiptune stuff that was out at the time. Anyway,pick up that Ys game, pop it in and just enjoy it for what it is. Then play the other versions, you'll find yourself comparing the soundtrack to the PCE version ALL THE TIME. That's how good and memorable the original score was. You'll also garner a bit of appreciation for these other versions just for how different they are to the PC Engine version of Ys I&II.
Back on topic, the PC Engine has many games that are great that aren't shooters. I don't believe comparing it to the Neo Geo is quite accurate in terms of a main console genre. The Neo Geo does have many fighting games, and with only 150 titles, it would seem to dominate the system. No surprise being an arcade-based console. The PC Engine however has maybe 60-70 shooters out of 650 titles, I wouldn't say the genre defines it though.
Some games I think you should try are the (and I'm surprised it has yet to be mentioned) Shubibinman series. The first title is an early but charming platformer and can be fun to play with little Japanese knowledge. The second game (known in America as Shockman), however, changed everything and for the better. The game has action and shooter levels making this a versatile, memorable title for the platform. The third game is much of the same but nothing amazing 'cept for some remixed tracks, which were awesome and some really great artwork. Another game you should try out is a Namco heavy-hitter by the name of Valkyrie no Densetsu (Legend of Valkyrie) released in 1990. It's very close to it's arcade predecessor... In any case, I am tired of writing but this list really does go on, it's all a matter of exploring the PC Engines amazing library of games. Part of the fun is the discovery of a great game and playing it to completion. When you do (and you will), you'll come here to talk about it.
