You could do a lot worse then Ikaruga as it's still one of the best shoot em ups out there. It was also a good reason to keep your gamecube around for the longest time until they ported it over to modern consoles. I know of a ton of shmups that are worthy of your time, just don't know if they are on steam which i guess is what you are after, so let's have a search on steam on wheter they are avaible or not. However i'm a console gamer and therefore i can't vouch how they perform on pc.
Crimzon Clover World Explosion -> Pretty recent release that has been recommended to me by a few people i know, since they were talking about it like the second coming of christ i decided to pick up the switch version. It runs pretty well, great bullet hell shooter that has multiple difficulty entries and is easy enough to pick up but hard to master, has some wacky mechanics i havn't seen in a shooter before. Yeah this seems like a good one to start imo. I don't know about the old school look as it looks more like a sega saturn game on steriods but you know old school enough.
Darius Burst Chronicle Saviours -> Latest entry in the long running darius series, you can't go wrong with this series, i'd say it's probably my favourite franchise in the entire shooter genre. All games are top notch quality, i played this on the Vita off all places had a good time with it. However if you ask my oppinion you are better off with the Darius Cozmic Collection that includes 7 games inc my favourite of the series the pc engine version of darius plus. I've also found this Hd version of G-Darius on steam but i can't say it's good or not, costs 30 bucks tough.
Ginga Force and Eschatos -> Going to list these together as they are made by the same company and are pretty much the same games, just the setting is different, had a lot of fun with these when they originally came out for the japanese 360. Story seems kinda wack from what i remember but overall worthy of your time and easy enough to get into. What suprises me even more the same company ported one of the rarest and best Bandai wonderswan games over to steam well what you know i learned something new today. i'm talking about Judgement Silversword offcourse.
Raiden V: Director's Cut -> latest entry in the longtime running series and it's is suprising how much raiden games are avaible on steam, yeah i meanly remember this 5th entry as it was an xbox one exclusive in japan for the longest time until they ported it over to other platforms. Still can't say much about this directors cut but as i only played the regulair version which is a decent enough entry in the series. Of all the raiden games i played the third game and fourth game are my favourites. There is also this compilation of four earlier games of which i only played the first raiden game of on the atari jaguar no less lol those were the days, i should hook that thing up and give it some love on of these days.
Bullet soul and Bullet Soul Infinite Burst -> Don't know why i waited this long since these should have been on top of the post, these are my absolute favourites in the bullet hell genre. Man they are perfect for beginners and aren't too brutal to get into and the gameplay is top notch, you feel like a badass in no time lol. Needless to say the music is top notch. I diden't even knew they were avaible on steam and were stuck in japan on the 360. Good to know so i can recommend them next time somebody asks the same question. I'd suggest picking up infinite burst over the original because it's essentialy an upgrade over the original game.
Obviously no shmup thread is complete without mentioning some Cave games as they are the masters in the bullet hell genre but i just don't want to take up anymore of your time. But it's always good to reminisce every once in a while about a genre that doesn't get the attention it deserves.
Hopefully this is a satisfying answer to your question Shagger and is what you are looking for. If you want to me to do a post of the above mentioned Cave we can arrange that aswell. It's also been fun to browse trough steam as it's a platform that i quite honestly never use.