why the "la"? does girlfriend/boyfriend not count as a close family member (and therefore it would not need a definite article)? marito/moglie does i think. at least i've "mio marito..." in earlier lessons.
In colloquial speech, "provare a vedere" can be used in a particular way so that it means something like "to check". For example "provo a vedere com'é la situazione" means "I'm going to check (how) the situation (is)"