The procedure for arriving and domestic-connecting passengers in Japan is the same. Everyone clears immigration and collects their bags for a customs check at their first entry point. Bags are only labelled as far as the first Japanese airport. After leaving immigration and customs in Haneda you will already be on the public side of the airport facing the exit to the street. You are in Japan now, you can walk out of the airport if you like, or you may turn right and follow the signs for domestic flights.
Since you are a connecting passenger it should be possible to check-in your bags for the domestic flight as soon as you arrive. This will save you from carrying them around. There are special check-in desks in the international arrivals hall that can handle this. Follow the signs for domestic flights: turn right out of arrivals and walk along the corridor to the far end. If this is not possible because you are too early (sometimes the desk is closed) then you can use the regular check in desks in the domestic terminal later.
There is bus over to the domestic terminal behind you when you are facing the bag check desk. The staff may invite you to use it. But do not use this, because it will take you through security and back into the airport.
There are airport lounges in the domestic terminal but they are not the place you want to spend more than a few minutes.