Visited Altyn Emel NP in 2022 while driving through Kazakhstan. It's most likely the easiest choice and on paper it also looks like a breeze from Almaty. Two problems with the location are: rental cars in Almaty all have a mileage limit because they DON'T want you to drive around the country with their cars so you'll need to look hard for a good deal that will give you unlimited mileage. In the end you are better off flying to the north for those sights and explore locally but even then the mileage limit of 100 km a day will be tight to do anything but just cruise around Almaty which is easily done by taxi anyway. Second problem is that if you don't have a car then you need to find some kind of driver but the tours (at least in 2022 which was still technically COVID time) there was zero infrastructure for this. You can try to find a guy who wants $100+ per day, and he probably won't speak English, but I got the feeling that nobody in Almaty really wants to do this as a job?! Before I start ranting, I eventually hired a car and did it on my own.

The park is only a couple of hours outside of town and once you are out of the city limits there aren't many cars on the road because it leads to...nowhere. I visited some weird rock carvings on the way which aren't inside the park but maybe it will give you something to do on the way. If you want a "fast" entry into the park you want to get off the A3 at Shengeldy and east will take you to the petroglyphs which at the time this site was tentative I figured was part of the inscription attempt because after all they have 3-4 sites around the country aiming for that but no, it's only natural now. Visiting these will take you a few hours mainly because driving on the small road is slow (you don't need a 4WD but definitely don't go with a low clearance Smart!) and I suggest you make it to the singing dunes before missing sunset. This was the best natural part I saw in the park because you won't spot mammals here and there. It also gets chilly cold at night so once you see the sunset you wanna move your butt out of the desert. I suppose this fits to the new inscription name although I don't find much special about them being just...natural deserts. The only nice  flora I found were these plans popping out of the desert sand. I spent the night at Kalilino (Ardak guesthouse) and started early the next morning, left the park at the northern side but once you are back on the A3 it's pretty much just driving for the rest of the trip (3h until you hit town traffic). In the end was a nice trip but wouldn't say one can find outstanding natural value here.