What's the best place to get Monster Claws early in the game?

You can get Monster Claws from Keese (bats) of any type. Just look for a place that has a lot of them and grind it out.

A place that comes to mind at that point in the game where you can find a lot would be the Waterfall Cave. That's the cave in Skyloft which is behind the waterfall where you had to go get your Loftwing. It should be relatively easy to go through that multiple times.

If you can wait a little bit longer, you might want to get the Treasure Medal before doing this to give them a better chance at appearing. You'll get this when you reach Eldin Volcano from one of the Goddess Cubes.

To the east of the Fire Temple behind the houses down by the lava.

The monsters in Skyloft (and the Waterfall Cave) only disappear after you've acquired all the Gratitude Crystals.

There are Keese on the Great Tree (close to the Bird Statue), and in the Volcano Summit, in the room to the right with the room to the dragon in the back.

Also, there's many Keese at the Pirate Stronghold - but only if you haven't beaten and opened the mouth yet.