The Baikal lake in one of the biggest non-salty lake in the world. Frozen during wintertime, it was around 2°C – 3°C at the beginning of June. If you want to swim, July and August are probably better but be prepared for a lot of mosquitoes :-). If you do not like them, June is probably the best month. Moreover the Olkhon Island is really crowdy in summertime and it can sometime take you hours just to take the ferryboat.
To go to the Olkhon island, the best way is to take a bus from Irkutsk. You can go to the main bus station and buy your ticket there. It is 1000 RUB for a one-way and there are buses at 9am or 2pm. You can also book online on https://avtovokzal-on-line.ru/ (but it is only in Russian).
It takes you around 5 hours to go with a 45min stop to eat and relax and a 10min cross by ferryboat.
Arrived on the Island you can camp (and ask the driver to leave you somewhere on the road) or go the main village (which is the bus terminus). You can also ask you driver to drop you directly in your hotel or guesthouse. All the drivers know the guesthouses and it is the best way to find yours (as there are no street names).
The mains activities on the Island are to go hiking, ride a bicycle, relax and visit the north of the island you can reach by bus. There are bus tours organized every day to the north of the Island (departure at around 9am). Just ask your guesthouse to book the tour. It is around 1000 RUB (meal included) and you will be back at 5pm. I went on this tour. As there are no roads, be prepared for a long uncomfortable hours in the bus :-). However it is the only way to see the landscape there if you don’t have a lot of time. The other way is to go hiking but you need several days I hadn’t (an to buy a permit as it is a natural park. You can buy the permit in the village).
On my second day on the island I went hiking in the forest near the village. A small river there was still frozen.
I finished my day by a walk on the cliffs next to the village and by long hours on the incredible beach, before going to the Banya, the traditional Russian sauna.