Fear not, your problem will likely be cured by your new battery. Hot starts are often where weak battery problems will show up first on a slant K, and as far as your battery being 'new', well maybe it was or maybe it wasn't, or maybe it was but had been mistreated (stored at low charge, etc.), or who knows what. First thing for the problem you described should always be a new battery when the condition of the old battery is unknown. The warning lights are probably just related to low voltage during and right after the start.
If the new battery doesn't resolve the problem then worry about other possible causes. But hopefully you won't need to go any further than a simple battery swap.