The possible declaration of independence by itself would not be the defining factor, what would be important is what happens after that; either:
the Spanish government prevails and it ends just being some guys "declaring" themselves to go to jail, and nothing else happens, or
the Generalitat government prevails and it becomes independent.
Now, since the first possibility does not change the status quo, let's focus on the second possibility.
The independentist parties have claimed that they will be part of the EU and that everything will be "business as usual" the following day, but the EU and some important members do not agree with that idea. At best, and providing that no member of the EU wants to block them, they would have to follow the admission procedures.
So, Catalonia is outside the EU, what happens now?
In relation to Schengen, it is outside. People from foreign countries allowed inside of Catalonia will have no right to travel through the EU1, unless the EU decides otherwise.
In relation to freedom of movement within the EU of Catalonian citizens, it will depend of their nationality. Spain allows double nationality, so current Spanish citizens could travel through the EU as Spanish nationals. Spanish nationals could even ask Spanish citizenship for their offspring born after the split. Of course, that depends on both Spain and Catalonia not forbidding such a double nationality, but I think that move would be rather impopular in both places.
Of course, by virtue of Catalonia being independent, they would be free to decide who they allow to enter, and the conditions of entry (visa, etc.). Since tourism is an important industry, it would make sense to keep the doors reasonably open to people from the EU, and the independentist politics have made claims to that effect.
In relation to trade the issue becomes way more complicated. Whatever their opinion, EU members trade is regulated by trade agreements that bind all of its members. Even if they wanted to, no member of the EU can just say "hey, let's open our borders" because it would mean opening the borders of all the states. So Catalonian trade would be subject to tariffs, import quotas2, and all the issues that, say, South Sudan3 trade to the EU has to face.
A trade agreement could be negotiated, but that requires plenty of time (as has been discused when talking about Brexit); and even countries that do not feel threatened by independentist movements may use the opportunity to block the Catalonian goods that are a competition to them.
Of course, the Catalonian government could open their side of the border, and unilaterally accept EU goods without limits or tariffs. But that could be troubling, as any producer in Catalonia could then move elsewhere in the EU and benefit from tariff-free trade with the EU and Catalonia, instead of facing tariffs and trade controls to access the EU market if it remained in Catalonia.
Remember, Schengen applies to people who are not nationals
of EU countries.
2And with no trade agreement assign Catalonia any of that quota.
3To mention another recently independiced country.