Getting to the beginning of the trail is not easy unless you go there by your own car.
After getting to Ust-Barguzin, pass the bridge over Barguzin River and turn left (almost backwards) at the first crossroad. In 5-10 minutes you will reach the National Park. From the checkpoint of the national park there is a dirt road. This is the only road, so there is no chance you will miss it. Just before the mountains, there will be a turn left. Take it, and in 1.5 km you'll see the informational sign at the beginning of the trail. Depending on your driving skills and condition of the road, the way to the campsite will take you from 1 to 1.5 hours.
By public transport:
First, you need to get to Ust-Barguzin. The only way to do it that does not cost you a lot is to take a bus from Ulan-Ude. The schedule can change every year, there is almost no source of up-to-date information. The only thing you can do is to call the bus services. They speak only Russian, so you will probably have to ask someone (for example the receprionist at your hostel) to call them and ask for the schedule and book the tickets. In summer the buses can be overbooked, so it is better to call in advance. Here is a website of the local bus station: http://avtovokzal03.ru/contact-us.html
Then, from Ust-Barguzin there are several ways to get to the beginning of the trail. First is to take a taxi directly to Svyatiy Nos cape. It's the most expensive, but also most comfortable way to get there. The cost will depend on the driver and your trading skills. Second way is to take a taxi to the beginning of National Park and then go by foot (40 km) or try to hitchike. It is the cheapest option, and quite easy in summer, when there are lots of people driving there.