If you are staying in the centre of Bishkek, this is an easily achievable day trip. Slightly less convenient than the equivalent option in Almaty, you are best off reaching the National Park by taxi or driving yourself, which should take around an hour and cost about £10 via Yandex.
I had rather low expectations in terms of what to expect from the park's infrastructure, but this was one of the most impressive and well-organised sites that I've visited.
Upon reaching the park, there is approximately a £2 entrance fee, payable by card at modern terminals with QR code turnstiles. Once inside, a small coach will take you up to the first set of trails and paths, with a further series of small golf buggy-style carts to ferry you further up to additional walking routes, all included as part of the entrance fee. At each of the small bus stops, there are small markets and coffee shops to pick up any goods you may need.
Once you have returned to the main entrance of the park, there are regular buses taking you back to the city centre, which are likely to be just as quick as waiting to call a taxi and cost only 50 som.
For specific route recommendations, take a look at the exercise section below.