Note that Korean names ending in Bong are 'peaks', San are 'mountains'
Insubong - (인수봉) : 810m in height is the second highest peak of the Bukhansan mountain proper, this peak has a prolific number of established routes (80+). Many are difficult slab/face routes with bolts and sections of aid (aid generally meaning bolt ladders). There are a select few moderate Trad routes centered around the main crack systems.
Seoninbong - (선인봉) : 708m in height. Technically this is a subpeak of the Dobongsan mountain. The other peaks however have less climbing routes. This peak is known for mostly difficult bolted slab routes.
Uiam - (우이암) : A single monolith out in the park. Has a few routes up it and has some more moderate options compared to the nearby Seoninbong.
Baekundae - (백운대) : 836m in height is the highest peak in the park and is the primary peak of Bukhansan proper. Less routes than the neighoboring Insubong, and often significantly crowded due to the walk-up trail that has been built on it.
Nojeokbong - (노적봉) : 716m in height, slightly south of Baekundae. Not as popular as the afrementioned areas, not as many routes or well developed beta.
There are many other peaks and rocks in the park, but from a climbers point of view, the above listed provide the best of this area.
Spring and Fall are the most popular and can be very busy. VERY busy. Weekdays are the quietest, saturdays aren't terrible, sundays can get extremely packed. On weekends take the first public tranist that opens around 530-600am, or if driving, try to be in the park by 7 at the latest.
Summer can be so hot and humid that it is not ideal to climb. Even if not raining, there will often be no climbers to be seen in July and August. Depending on where you come from and how acclimated you are to hot humid climates, you may find yourself quite dehydrated by the time you even finish the approach, and will be battling sweaty hands the entire climb. There is little to no water resupply once you leave the trailhead. On the other hand, if you are used to climbing in a tropical jungle like climate, you will find this season the least crowded!
Winter is very cold and you can expect a high probability of verglas on the mountain if it has been precipitating, as well as snow possible at all altitudes. Hiking is still popular in winter along the maintained trails. If conditions are right however, I would expect that all the routes are still climbable. Haven't tried myself yet. Maybe some crampons and a tool would be adviseable just in case some sections present mixed terrain.
You can get food and water at the Do-Son-Sa entrance. There is a hut located near Baekundae that you can reserve a bunk in, if you are doing multiple days in a row. It also has water and serves limited food. Dobongdong is likely similar, with many small shops and restaurants leading up to the trailhead. Ui-Dong has a variety of outdoor clothing and equipment stores.
Transportation is very easy for this park. On the east side of the park is Ui-Dong (우이동), slightly northeast is Dobongdong (도봉동). This is the best end-point for public transportation. I'd recommend using Daum Maps app on your phone to map out the most efficient route from your location to the mountain, no single bus/subway route will be easiest for everyone.
Baekundae/Insubong/Nojeokbong: From Ui-Dong, grab a taxi to Do-Son-Sa (도선사). This will reduce the walking to the minimum. You can take a temple bus that runs every 30 minutes, but for a $5-ish a taxi ride, why wait?
Uiam: Begin walking north from Ui-Dong. Alternatively, use the Daum Maps app to route yourself to Dobongdong (도봉동) and head due west from there.
Seoninbong/Dobongsan: Route yourself to Dobongdong (도봉동) and head due west from there.
Hardly any. No camping overnight and make every effort/plan to be walking out of the park at sundown. It is technically illegal to be in the park after end of evening nautical twilight. It does happen and it isn't harshly enforced, but it is the right thing to do barring accidents/emergencies that delay you. Park entrance fees have been on/off in the past, just ask around at the trailheads to get the latest on fee regulations.
Illegal. Can stay at the bunkhouse at the hut near Baekundae if so desired.