To make a long history lesson short, the bulk of the collection housed at the palace was brought over from China during the civil war. There was a limit to how much the army could take with them as they bounced around the country, so only those objects deemed truly valuable were brought along. As a result, the Taiwanese half of the museum - the other is in the Forbidden City in Beijing - houses a disproportionately high volume of culturally meaningful art. Thankfully, the curators at the museum have packed the permanent collection with great pieces, and there are which come out of storage in rotation.
With all the great pieces of art from China's past on display the museum gets quite busy, especially on the weekends. Because this is a must see, every tour group which comes through Taipei will make a stop here. If you want to be able to enjoy the museum in a less frantic environment, shoot for Mondays to Thursdays. Every other museum is shut on Mondays, so there is a bit more traffic, but you'll be avoiding the tour groups which swarm the building Friday through Sunday. If you have to come on the weekends, get here early, or during lunch time because the tour groups will be gone. Avoid coming in the mid afternoon on the weekends.
Buses, taxis, and tourists crowd the street level entrance on weekends.
Sadly, the museum strongly enforces the no photography rule in its galleries, so there isn't much to post here. Instead, satiate your pre-visit curiosity with this rough guide for how to see the best stuff.
Nearly everything that will capture your imagination and interest is on the third floor. It's best to get in and head straight up to the third floor where you'll be politely filtered through a set of U-turns to see the Jade Cabbage. In nearly a dozen visits to this museum this area has never been empty because the little sculpture is a outlandishly famous object in Chinese culture. The cabbage is no doubt the main attraction and sometimes there is a subtle pushing match that happens around the cabbage. The other display case in this room has a rock which looks shockingly like freshly cooked meat which often impresses more than the cabbage does, and a tiny pair of dancing bears. As of October 2015 the room had been designed to filter people through in single file while quickly giving you unobstructed access to each of the cases.
Once you've seen the two main rooms on the third floor you can decide what else interests you. Your next stop should be to see the ivory carvings which are currently housed on the first floor. They used to be on the third floor in an exhibit with tons of reliefs carved into rhinoceros horn, but the gallery has moved, likely in an attempt to spread crowds more evenly in the museum. It is currently being called "precious crafts" on the map, in hall 106 with a picture of a strange ball. The ball is what appears to be a series of ivory rings nested inside one another and it hangs as part of a giant ivory mobile. Everything in hall 106 is a masterpiece, but much of the treasures here are made from animal bones. For those with moral concerns there is signage to help visitors understand the pieces on display in the context of current conservation programs.
Also on the first floor is a hall full of religious relics (halls 105 - 107) which is worth at least a quick look. There is a massive pagoda which features hanging bells - that really do hang - as well as tons of finely wrought gold crafted in reverence to the Buddha.
After seeing floor one you can consider heading back up to the third floor where the museum has an interactive exhibit about bronzes. The old bronze bells and urns which are spread around the museum are not very impressive by modern standards, but after a tour through hall 300 you'll be ready to at least appreciate something about halls 301, and 305 - 307. There are some very good bilingual displays about the evolution of Chinese characters which are accessible to even non-Mandarin speakers. The jades in halls 306 - 308 are also quite impressive given their context, but are nowhere near as well loved as the jade cabbage you've already seen.
The second floor should be the last stop for visitors who have already taken tons of stops into Chinese museums, and it should only be explored once you've exhausted the possibilities on the third floor. The middle level has a collection of ceramic wear, antiques, calligraphy, and paintings. Foreign visitors usually find the galleries somewhat inaccessible due to language and cultural differences, so save it for the end. It's definitely worth a visit if you're into calligraphy - many bearing the stamps of the emperors who owned them - and brush paintings, but for most people it quickly becomes a slow walking tour just to say you've seen it.
You'll probably be exhausted after all the slow walking and stooping common to museum going, so head outside to the Zhishan Garden which is free with your museum ticket. The entrance is all the way down the stone steps around a stone wall on the left hand side as you finish the final descent to street level. There are a few drink vending machines inside, and you're welcome to take pictures here in the often deserted garden.
The museum has raised its prices to 250NT for a standard visit. Students with valid foreign ID (ISIC) are 150NT, while those studying full time in Taiwan - language schools excluded - and carrying their ID are let in free.
Special exhibitions regularly set up shop in the second exhibition hall which is on the upper level on the left side when facing the museum's main building. After visiting a few of these exhibitions the only unbiased review would be to say that the quality and desirability of these exhibits varies greatly - despite commanding high ticket prices - and unless you've been living in Taiwan for a while you're likely better off not going inside.Getting to the museum is done by bus or taxi, usually from an MRT station. You should take the MRT to Shilin station of the red line - now called line 2 - or to Dazhi station on what is now called line 1. Shilin offers more buses with higher frequency than Dazhi, but Dazhi has a slightly lower taxi fare. Once you arrive at the station there are a set of local buses which will take you to the museum proper or you can grab a cab for around 120NT. Ask the staff in the information counter at the MRT which buses are most convenient at that time of day. Everyone who works in the MRT is able to speak some English, so get help when you need it.
The stairs to hall 2 are next to the big white gate