From Moab take 191 north to the left hand turn onto 313 which is signed for Dead Horse Point and Canyonlands NP. Follow this to the entrance station which is about 32 miles(40min). Immediately after passing through the entrance station, take a left turn onto the Shafer Trail and follow this for about 30 miles(3 hours) to Monument Basin. The White Rim road is a 4wd only road, and requires either a day permit(free online) or if staying overnight a backcountry camping permit(needed months in advance). 

Alternatively, you can hike from Island in the Sky via the not well known Government Trail. From the Grand View Point overlook trail, hike along the tourist trail for about 10-15 min. Leave the trail to the left and find a 3rd class chimney which leads to some cairns and a 4th/5th class downclimb through the cliff bands. (this is not signed or maintained by the park). After reaching the talus below, continue down towards the basin and navigate the smaller cliffband below, either through finding a 5th class downclimb or traversing hard right(south) towards the drainage. This should take you to the White rim road.

Once you've reached Monument basin on the White rim Rd, access down into the basin proper is done in one of three standard ways.

1. The most common is the single rappel approach on the west side of the basin, which is fastest for Standing rock. See Standing rock page for more details.

2. There is a 4th class entrance on the north side of the basin, which is useful if accessing towers closer to that end(Staggering rock)

3. There is a 3rd class gully entrance on the south end of the basin, which could be useful for towers like Pixie stick.

All these approaches require sniffing around and looking for cairns/previous passage.