From Gunnison, drive up and over Monarch Pass on US50 and down to the small resort town of Garfield. Continue east on US50 from Garfield for another 4.8 miles to the turnoff for CR224 on the left (north) side of the highway. Coordinates are: N 38° 32' 30.94" W 106° 13' 01.72".
From Poncha Springs, drive west on US50. At the small community of Maysville, measure from the intersection of CR240 (this is the road taken to access the Shavano Campground and the one of the climbing routes for Tabeguache & Shavano.) and US50 and drive another 1.6 miles west to the turnoff for CR224 on the right (north) side of the highway. Same coordinates as above. Elevation 8,515 ft.
Once on CR224, drive generally north for 1.5 miles to where CR224 makes a sharp turn to the left and becomes CR228. Continue for several miles on CR228 which follows along and above Cree Creek. At 6.5 miles appx., CR228 will come to an old marble quarry on the right. Elevation here is 11,680 ft. That will keep Taylor & Aetna a sporting hike. Park here if you do not have 4WD with good clearance. If you do have one, you can continue up CR228 another three-quarter mile to an elevation of just above 12,000 located on a broad eastern ridge of Taylor. Park in this vicinity to begin your hike.
Alternate Approach: On US50, just a little east of Garfield, turn north onto CR228 at these coordinates: N 38° 33' 02.63" W 106° 17' 10.53". The road follows up Taylor Gulch and passes the "Garfield Mine." At 3.1 miles, the now much rougher road leaves the gulch and switchbacks four times to the west, then makes a long contouring ascent to the same broad, eastern ridge of Taylor mentioned above where you can park at about 12,000 ft. While the overall mileage of this access is shorter, it will not save much time because of the slower driving on the 4WD road.
The best National Forest CG we can recommend is the Monarch Park CG, the turnoff of which is located 3.2 miles west of Garfield on US50. CR233 turns off on the south side of the highway and descends to cross the creek, then continues south to the campground. Where the road crosses the creek, there are some primitive sites there if the campground is full. Our last visit there, they were doing beetle-kill mitigation and so some sites were closed. Overall, this is a nice location. Coordinates for the turnoff from US50 are: N 38° 31' 43.46" W 106° 19' 31.69". Campground coordinates are below. Here's a link to the San Isabel NF report on that campground: https://www.fs.usda.gov/recarea/psicc/recarea/?recid=12677
For primitive camping closer to the TH, where CR228 reverses direction at the old marble quarry, there appears to be some primitive camping in that location, and if you drive up from there to the SW, the road passes some old cabin structures where there also appears to be some primitive sites.