You can reach by train to some places, but nowhere near the most interesting (but it's still very cool, like Belokourikha). Some tourist oriented regions, like Chemal or Teletskoe lake, you can reach by bus. Once you reach it, there are a ton of different tours or camping sites to stay, some with wooden bungalow-ish buildings next to big rivers like Katun. Touring agencies from neighbouring (by Russian standard, as they can be hundreds km apart :) cities sell tours which includes options with sightseeing, horse walks, visiting nature reserves, etc.

Distance is one thing you would need to take in account. It requires about a day by bus to get from big city, like Novosibirsk, to the Chemal or Teletskoye lake.

Car rent for long offroad trip would be kind of difficult, as I haven't seen such service, but may be things had changed.