Let’s start with getting the items
Wood pallets are easy to find - you can even browse the internet. Or you can go ask a local supermarket if they are selling some. Have in mind that you need good quality ones - nothing suspicious, soggy, stinky or with odd colours.
LED strip lights could be bought on the internet for around $12 or £11. You don’t need hundreds of metres - just 5 would be enough.
Sander - if you don’t have it, it’s not a big deal. Use sandpaper - cheaper option, but you will have to work a bit harder to make the planks smoother. It’s not that essential to sand the pallets, but it will definitely make them look better and it would be easier for you to apply the paint.
You don’t have to paint it, but it will definitely look better. Unless you want the bed to look more natural.
Plywood - get it from your local home improvement and construction stores. The bigger brands could sell them on really good prices. You don’t need much - just to cover the surface.
Carpet - a finishing touch, but it’s not essential. You can cover it differently if you want.