If I move Traveling Merchant and discard Flame Chompers, do I finish the Traveling Merchant draw first or do I pay a power for Flame Chompers immediately and summon it?

You finish the Traveling Merchant trigger (discard then draw) completely before Flame Chompers' trigger ever goes on the chain. You cannot play Flame Chompers mid-resolution.

Here's the exact sequence:

1. Traveling Merchant moves — its "When I move" trigger goes on the chain.
2. Chain resolves — you discard 1 (Flame Chompers), then draw 1. No interruptions happen here; you cannot interrupt a resolving effect (Rule 354.3).
3. After that trigger fully resolves, discarding Flame Chompers creates a new triggered ability that goes on the chain.
4. That trigger resolves — now you may pay the fury power to play Flame Chompers.

So the draw happens first during step 2. Flame Chompers' play opportunity comes after the full discard-and-draw is done (step 4).