It's a square taper crank on a nutted bottom bracket spindle. You'll be able to see the square taper interface once you remove the nut. The body of the crank puller gets threaded fully into the crank arm threads, and then the puller's inner driver pushes against the spindle while pulling the crank arm off.

Replacing the crank with a triple will probably require replacing your bottom bracket (unless both cranks have the same type of interface and require the same length spindle). Have you planned on that? It will take more specialized tools. Your square taper bottom bracket requires different tools than the common cartridge square taper bottom brackets of recent years. So if you do end up replacing your bottom bracket with something more modern, then that's probably going to take yet another tool. If you're a DIY guy, then it's more reason to buy tools. Otherwise, you might just let your local shop do the installation. Just throwing this out in case you weren't aware how changing one component can result in changing three components and buying six new tools.