For the lizard, once you arrive in the Goron town, visit the armory. They sell a full set of armor that will take care of you while you're there.
Now for some annoying reason, that same armor doesn't work in the desert.

There's also the cheaper option of elixirs and food stuffs. I don't remember recipes off the top of my head, but I know I found them when I leading up to the area.
Hope that helps!