Basically all of the solid and the liquid blocks, but they can't climb single slabs but they can climb double slabs.
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Yes, spiders can climb almost all blocks. Use minecraftwiki.net if you have any more questions ;)
Minecraft characters are approximately 2 blocks high.
You can't.
you get carrots in minecraft from baby zombies on chickens or spiders, you have to kill them and then you will get them.
you have to kill spiders