April - May should have decent weather. Not cold but not too hot either. It won't be super humid yet but Smog coming from China might be a problem. Itaewon is great for a variety of food, Hondae is good as well. Make sure you try this little bread thing with an egg in the middle of it, I don't know what's it called but it was so good! I wish i had eaten more of them.
I would recommend visiting N. Seoul Tower later afternoon towards nighttime so you can get that awesome night view. You can also buy locks and write your name on them with your girlfriend and lock them onto the viewing platform, you'll see what I mean when you visit it. I would also recommend taking the gondola, it's cheap and you won't have to walk 2 miles to go all the way to the tower.
A lot of palaces also offer traditional costumes that you can rent out if you or your gf are into that sort of thing. It's pretty cool walking around the palaces in traditional clothes and it's fairly cheap.
Visit Lotte Tower's top glass floors during a clear day (minimal smog) to get an awesome view of the entire city.
Make sure you get a T-money card for public transportation, they work with trains, buses and taxis. You can get them almost anywhere and depending on their promos you can get some cool looking ones (I have a pokemon charmander card)
You can also try watching a 4D movie if you haven't tried it yet, I'm sure Avengers will be awesome in 4D.
Saunas are pretty cheap as well, I went to the Silloam Sauna and had a good time for just $11 or about 12,000 won. There's all sorts of bath options and different types of dry saunas that you can try out there.
Taxis are great but the subway system is amazing, don't worry it's easy to understand and use.
I spent a year in Korea while I was stationed there in 2017, I had a blast, sometimes I wanna go back.
