How do you play terraria without downloading it?

Answer 1

Technically impossible. Without any way of connecting your GPU, CPU, RAM, and other proccessing devices to all of the data, you can't render it on your screen.

Unless of course you're using a cloud system, which for a simple game like Terraria may work.

Answer 2

There are 'wrong' ways to play Terraria but there isn't really a 'right' way. But the first thing you probably will want to do is build a house out of wood so that when night comes you don't get massacred by Zombies and demon eyes. Then you can start digging with your pickaxe to find underground metal ores and treasure to upgrade your tools and armour, and build more houses to get people to move into them.

You can ask the character who spawns next to you (the Guide) what to do next if you're not sure. He also gives you crafting recipes.