Okay, sorry for the day late response. But here we are.
So the strips I'm currently looking at are these. Now I havent finished my research on them, but I know the company sells little controllers for these. Basically like a tiny box that you can program the strips with.
https://www.amazon.com/dp/B01CDTEKAG/ref=cm_sw_r_cp_apa_i_fDRrFbB6M84QQ
Otherwise, any regular LED strip usually has about 16 colors with a remote. Some are sound activated, some arent etc. You just have to shop for what your desired effects are. The best place for recommendations if you dont wanna do research are the ones pinned at the top of this subreddit.
Now Phillips hue, which I'll provide a link for as well, are one of my favorites. They offer over 16 million colors to choose from and should be operable via your phone. The downside is that the strips are shorter in length and require extensions and a hub. Basically the core stip can have a certain amount of extensions(cant remember if it was two or three), but that means you'd purchase, depending on the size of your room, 2 to 3 regular strips, plus however many extensions, and a hub. Without the hub you cant control the strips. Sucks, but tbh I think the amount you can do with these makes it worth it to put up with purchasing a hub.
Lightstip base
https://www.amazon.com/dp/B0167H33DU/ref=cm_sw_r_cp_apa_i_VKRrFb4E4CQVV
Extension
https://www.amazon.com/dp/B0167H31CI/ref=cm_sw_r_cp_apa_i_XLRrFbV0VK905
Hub
https://www.amazon.com/dp/B016H0QZ7I/ref=cm_sw_r_cp_apa_i_xMRrFb44C8D80
*Note: These are all just Amazon links to the individual products. Sometimes you can find Phillip's starter kits that include the hub, most common being with their LED light bulbs set. So depending on how much your budget is and how much you like the product, you can try to get more for a cheaper price via these kits and places like ebay/offerup/mercari.
Now If were talking laser light. I'll send a link to the one I'm looking at myself, but these you really should put your own time into researching. There are a lot of different models and brands that do a array of different projections.
This is the one I'm looking at on amazon right now, but I'm still not set on it.
https://www.amazon.com/dp/B07195BNXF/ref=cm_sw_r_cp_apa_i_gRRrFbHF8JQ4S
Lastly, just as a extra addition. I bought this lovely "laser" and wave projector light. You can definitely find better ones on amazon, but I heard this one was silent running so I immediately grabbed it. I love it, but it has two issues for me. The laser "pulse" setting, where its supposed to slowly fade in and out like stars is choppy. Meaning it fades a little, stops, little more, stops, off, then on a little, etc. Second, the bluetooth sucks. It's not loud AT ALL. But the sound activated mode is super nice imo. Just turn it the lowest u can that it still functions, and play the real music louder somewhere else. Easy.
Here is the link to mine. However, amazon sells SOO many copys of this for cheaper. Just take some time to look through the features, customer reviews, and most importantly, customer videos/photos.
https://galaxylamps.co/products/galaxy-projector
Hope this super long nightmare of a comment helps your adventure to a wonderful cave!