Papaya salad, som tum, is probably my favorite Thai dish. It's made with green (unripe) papaya, which is the number one key - it just wouldn't be the same thing with ripe papaya. Unripe mango would work if for some reason that's easier for you to find than unripe papaya.
The tricky thing is shredding it. I've tried a cheese grater in the past and wasn't happy with the results, so I tried the Thai method - making lots of cuts all over the peeled papaya with a sharp knife and then slicing them into shreds. It's trickier than it looks on youtube - unripe papaya is a lot firmer than ripe papaya, for one thing, but there's also the cavity in the center, so your slices may suddenly come detached before you intend them to.
My papaya came out coarser than you'll find in a Thai restaurant. The texture/taste difference is noticeable, unfortunately! (Or unforts, as the kids say.) I "fixed" this by taking the largest pieces and putting them aside to make green papaya kimchi.
Traditionally you serve papaya salad with raw tomato and long beans. I don't have any long beans and I'm not about to buy fresh tomatoes in Februfuckinary, so. The dressing is simple - sugar, lime juice, fish sauce/dried shrimp/salted crab/something along those lines, garlic, and Thai chiles. Somehow I don't have any fresh chiles, so I used pickled green peppercorns instead, mainly because I'd been looking for an excuse to use them in something.