Ignore the "connecting passenger" signs, they refer to people connecting to flights outside Chile, i.e. go direct to immigration.
Citizens of the EU, Brazil, and (recently) USA: ignore the queues to pay entry fees.
The queue for immigration and/or customs can be long (over an hour), you might want to go to the loo before joining it.
The arrivals form doesn't ask for the flight number, but the immigration officer will probably ask you for it, so have that handy when you get to passport control.
The immigration officer will give you a small paper receipt. You need to keep this because they'll expect to see it when you leave.
After you get your bag, you hand in the customs form, go through customs (this includes a bag scan and possible search). Make sure you tick the X saying you are bringing food unless you are 100% sure you have nothing with you, including snacks you brought for the flight.
After leaving customs, you enter a crazy hall jammed packed with people meeting relatives or touting for taxi business.
If you are going to La Serena the same day: barge through the crowds and head for the glass lifts (only 2 for the whole airport, so its a bit chaotic).
You might want to get out some Chilean money at the airport (there are machines in arrivals and departures). Most places take cards, but you need cash for taxis. Make sure you warn your bank that you are traveling, otherwise they might stop your card (credit cards also)!
Take the glass lifts upstairs (it is only signed for international departures, but it is for both). Turn right and walk almost the whole way along the hall to LAN domestic check-in. You might already have a boarding pass and a luggage tag, so all they will do is take the bag off you. Or you might need to check in at a electronic podium before doing bad drop. Or you might need to go to one of the desks on the opposite side to the bag drop to check with a human. Go with the flow, eventually someone will direct you to the right place!
Once you've dropped off your bag and got your boarding pass you continue to domestic departures, it's all well labelled.
In domestic departures there are quite a few cafe's with wifi, but you might have to ask for the password. It is a good idea to eat in Santiago, if you have time, because you will be taken directly from La Serena airport to the AURA compound (Recinto) and there are no cafe's there.
Assuming you have told Ximena at Aura your travel plans, you will be collected at the airport by the AURA taxi driver (the sign will say AURA not CTIO) and taken to your chosen location (Ricento to take a shuttle; Ricento to stay overnight; hotel)
If your bag does not arrive the AURA taxi driver will help you sort out lost luggage forms. But it likely will take several days to get to you if it needs to be sent up to the mountain after you leave La Serena -- so take spare underwear and a t-shirt in hand baggage.