The closest airport is in Guìlín. Your hotel should be able to organise taxi rides directly to the airport (about ¥240, one hour).
Direct bus links:
Guìlín ¥20, one hour, every 15 to 20 minutes (6.45am to 8.30pm), leaves when full
Guìlín Airport ¥70, 1½ hours, 8am, 10am, noon, 1.30pm, 3.30pm, 5.30pm, 7pm and 8.30pm
Nánníng ¥160, six hours, three daily (8.40am, 11.30am and 3.30pm)
Shēnzhèn ¥150 to ¥180, eight hours, seven daily (1.30am, 12.30pm, 1.30pm, 3pm, 8.30pm, 9.30pm and 10.30pm)
Xìngpíng ¥8, one hour, every 15 minutes (6.30am to 6pm)
Yángdī ¥9.50, one hour, every 20 minutes (7am to 6pm)
The bus from Guìlín to Huángyáo (¥61, 9am and 1.30pm) only stops in Yángshuò erratically, so check before purchasing tickets.
Yángshuò town itself has no train station, but there is a station called Yángshuò station servicing high-speed trains in Fànzèng Shān (饭甑山), near Xìngpíng, 14km away. Regular buses (¥20) connect the train station with Yángshuò, travelling via Xìngpíng. Trains include:
Guǎngzhōu 1st/2nd class ¥141/117, 2½ hours, five daily, 8.21am to 6.49pm
Guìlín 1st/2nd class ¥25/21, 30 minutes, seven daily, 9.03am to 7.19pm
Nánníng 1st/2nd class ¥157/131, 3½ hours, one daily, 9.03am