This is the easiest route up Tuku in winter, and will be described as a winter route. It can also be done in summer. You will need to read the topo map; there is no trail.
From the Burger King in Moab, drive SW along Hwy 191 7.4 miles to a junction with the La Sal Loop Road. Drive up the La Sal Loop road for 12 miles, then take a dirt road on the right towards Geyser Pass. Drive on this for 5.55 miles and take a right onto a smaller dirt road. Follow this another 1.65 miles to the end. In winter, the road is plowed to a plowed parking lot just before the junction with the Gold Basin Road.
A topo map showing both routes to the top of Tukhunikivatz from Gold Basin: 1-Gold Basin. 2-Alternative Gold Basin Route.
Ski or snowshoe the Gold Basin Road a short distance to the NNE spur of peak 12,048 on the topo map. Begin hiking on the small trail (not visible in winter) that spurs off from the end of the dirt road. Follow the trail downhill, but as it turns to the left, continue walking downhill in the clearing to the stream. Ford the stream and walk up the clearing heading southeast towards peak 12048. From here, continue climbing up (this is a talus field in summer) and scramble up a steep slope ( loose talus and scree in summer) to gain the ridgeline. (There is usually some thin patches of snow that run up most of the slope well into summer, and these make the route much easier). The route climbs rather steeply to peak 12,048. This will be the most difficult part of the climb. You will need snowshoes and an ice axe.
Follow this ridge to the summit of Tuku. The ridge will have good hard snow for cramponing from peak 12,048 to Tuku. The ridge over Razor Fang to Peale can also be done, but this is not an easy route, and it is difficult for most people to climb both peaks in a day (in winter). There are also other routes of Tuku from Gold Basin, but they are more prone to avalanches. Enjoy, it's beautiful in winter (and in summer too!).
Gold Basin, looking east.
Ice axe, crampons, snowshoes or skiis in winter.
In summer take a good pair of boots.