salir vs quedar
Quedar is making the plan to meet. Salir is the activity of going out. You can have one without the other.
salir
Going out — leaving the house, doing an activity, or dating someone.
Salí con mis amigos anoche.
I went out with my friends last night.
The activity — you went somewhere together. The evening happened.
¿Sales con alguien?
Are you seeing someone? / Are you dating anyone?
'Salir con alguien' also means to date. Context usually makes it clear, but worth knowing.
No me apetece salir esta noche.
I don't feel like going out tonight.
Salir = leaving the house. Quedar could still happen at home.
quedar
Arranging to meet — the plan, the agreement, the arrangement. Common in Spain.
¿Quedamos el viernes?
Shall we meet up Friday? / Are we on for Friday?
The arrangement itself. You could then go anywhere. Quedar is the plan, not the outing.
Quedé con ella pero al final no salimos.
We arranged to meet but we didn't end up going out.
Quedar (plan) without salir (activity). They're two separate things.
Quedamos en el bar de siempre.
We arranged to meet at the usual bar.
Quedar + en = where you're meeting. Very natural, very common in Spain.
Tricky bit
In Latin America, 'quedar' for arranging to meet is less common — they'd say 'juntarse' or 'encontrarse.' If you're in Mexico saying '¿quedamos?' people might look confused.