A while back a friend planning a visa run to Korea asked me for must-do's while there. I of course said I'd be happy to provide a list, but soon realized it'd have to be limited to Seoul. And after that followed the insight that I only really know what to do in Hongdae, so I put together a few suggestions on activities towards the full Hongdae experience. Keep in mind that I've only ever gone to Korea on vacation on a few occasions, so this list is far from complete - you are more than welcome to leave your own suggestions in the comments!
First of all, you're going to need some place to stay. Having friends to stay with is great - especially if they work so much staying at their place is the only way to actually meet them - but my most heartily recommendation would be to check in at the Hongdae Guesthouse 2.0 - Yellow Submarine. For a sum less than a nightly taxi fare to the outskirts of Seoul (not to mention saving the trouble of getting a taxi at that hour as a foreigner - especially one that'll actually consider taking you there) you get a bed to stay in close to Hongdae, a mere 5 minute walk from all the noise which as far as I'm concerned is just perfect. There's internet and breakfast, but most importantly the people running the place are fantastic human beings. The guests I've encountered there have all been great too, so even if your plans are canceled last-minute you can be sure to have a great time just hanging out with the hostel people. I'd pick the Yellow Submarine over a fancy hotel any time, and even now when my situation has changed so that I couldn't stay there myself I drop by just to say hi. This has nothing to do with Korean indie music but is the most valuable tip I can provide.
You will also need somewhere to eat. Korea in general - Hongdae being far from an exception - offers plenty of opportunities to do so at virtually every hour of the day. A surprising amount of people related to the indie scene are running their own restaurants or cafés and some time I might attempt gathering a list of them all, but for now let me just recommend my favorite place for fried chicken: Reggae Chicken! Run by Ryu Kwanghee of The Mustangs and and Nenoon fame. There are currently two locations, the original near Hongdae station and the new one by Sunshine, close to Sangsu station. Perfectly crisp chicken with delicious Jamaica inspired flavor served in a reggae infused environment.
For snacking, there's Takomong, close to Sangsu station, a popular hangout especially among hardcore band members. It is run by Seo Jung Ki of God of Emptiness, and he serves delicious Japanese style balls made out of octopus.
Heading for coffee one might be tempted to seek out the actual first shop of Coffee Prince, as seen in the drama with the same name. I would strongly advice against it, having had an experience not to far from the one described here. Instead, look down the street to the other side of the road for Coffee Lab. Doesn't have the same connection to either drama or lovely indie music, but coffee is much better and your experience guaranteed to be better.
As for activities, being an indie music fan in Hongdae you absolutely have to go to a show. No excuses. Use Korea Gig Guide or check out IndieNews_bot on Twitter (if you can read hangul) to find out which shows are on and if necessary consult a map - either the one at Korea Gig Gudie or the one above. My own favorite venues are Live Club Ssam (unfortunately no longer in existence) and Club SPOT, but of course if there's a band you want to see playing somewhere that's where you should go. Personally I try to avoid places where expats flock (e.g. Club FF) - not because I dislike the company of expats (not only are some of my favorite Seoul people expats, but I'm merely a tourist myself) but because I enjoy the atmosphere so much more elsewhere. Going somewhere as the only foreigner you're very likely to draw some attention towards yourself, but on the plus side it also makes it very easy to make new friends.
For the girls there is one more activity that deserves a mention - bellydancing! I'm not much of a dancer myself, but having been mesmerized by Eshe's dancing while watching Apollo 18's Orbis music video I wanted to at least give it a try. Eshe, being the truely lovely person that she is, is happy to share her dance with any students regardless of whether you just want to give bellydancing a try during a short visit or plan on hanging around long enough to return for more classes. I attended a couple of Eshe's classes during my first visit to Korea and though a mere beginner making mistakes in everything I had a great time and felt truly welcomed by all of her lovely students. Here's the summer schedule and a map to the studio.
Eshe in action!