Know Before You Go
The lake is located very close to the city of Almaty, and can easily be visited as a half-day trip from the city. The easiest way to get there is by taxi. The fare there and back should be around 6000 KZT. If you order a taxi on a phone app, have it wait for you if you're not planning to be there all day. It won't be that much extra. You can also have your taxi stop up or down the way on the mountain to get photos on the gorgeous drive.
Another option for getting to and from Big Almaty Lake is to hitchhike, although this should be done at your own risk. Visiting the lake itself is free of charge, but you must pay a fee to enter the Ili-Alatau park at the bottom of the mountain. The fee is a little over $1 USD.