Can you really get warts from toads?

No!

Warts are caused by a virus, called human papilloma virus (HPV) that replicates in the basal layer of the epidermis. The virus causes a weird cauliflower-like growth from infected epidermal cells.

Toads don't get this virus (hence the "human" part of the virus' name!), and they can't spread it.

I'm not sure where the myth comes from - some people claim that it's because some toads have raised, coloured camouflage patches that look like warts. The story sounds plausible enough, I suppose...