Travel to Sumatra. The main entry-point for Sumatra is its largest city, Medan. This is great for getting to Mount Sibayak, as both are located in the northern part of the island. If you are flying, then you can reach Medan either by international connection in Kuala Lumpur or Singapore, or by domestic connection via Jakarta. Another option is to take the sea ferry from Penang, Malaysia.
Get to Berastagi. Once you are in Medan, you need to travel to Berastagi, the town below Mount Sibayak. The two means are by taxi or public bus, as there are very few mini-buses plying the Medan-Berastagi route. The public bus will be cheap but crowded. The hired taxi and driver will be relatively expensive, but quicker and more comfortable. Which one you take depends on your budget and sense of adventure. If you are already in Sumatra, there is a mini-bus route from a small town on the banks of Lake Toba that goes to Berastagi on back country roads. The scenery is spectacular and the trip is terrifying, but it saves you doubling back to Medan.
Hike Mount Sibayak. You do not need a guide to do the trip. Simply catch a local bemo (mini-bus converted to public transit) out to the park and your guesthouse or hotel will tell you how. Once there, walk up using the paved, good road that will stop near a trail and leave you with about half an hour's walk to go. While the road is good, the ascent is steep and there will likely be no one else out there except you (and your party), so bring water and snacks. The whole trip should take 2 to 4 hours to the top (2,212 meters). The final approach and the crater itself are like an amateur volcano fan's dream. While there is no lava, there are fumaroles, creeks with simmering greenish water and a staggering crater sitting among the clouds. You can either go back the way you came or find the trail on the other side of the crater that will take you down through the jungle.
Visit the local hot springs. On the other side of Mount Sibayak is a small village featuring a pair of spas taking advantage of the local hot mineral springs. Bring your swim trunks and relax after the hike with a dip in the waters.
Don't bother with the local Karo villages. While there is some merit to them, Berastagi is poor and semi-isolated, and a visit to these villages will be more depressing because of the poverty than anything else. If you are in Sumatra at all, you will likely go to Lake Toba and the Batak peoples, and these are a better people to explore local cultures with.
Try beetle nuts. If you don't mind tobacco, give it a try. You roll up the nuts with tobacco and a lime pasted into a leaf. It has quite a kick and a mild narcotic effect. Other local treats include babi panggang, a pork dish, and the local alcohol called tuak.
Add Mount Sinabung into the bargain if you are feeling adventurous. Sibayak's neighbor is a much harder climb and higher at 2,451 meters.