Pan out means to happen or develop in a particular way. So the sentence, Let’s wait and see how the day pans out before we decide what to do tonight means, “let’s wait and see what happens today before we decide”; that is, “let’s what and see how the day develops before deciding”.
I didn’t know how it would all pan out when I bought the ticket to France… I thought it would be a lovely holiday, but it was a disaster.
Things don’t always pan out how you want them to. I started the business but after a year I had to give up because it wasn’t working.
I don’t think this will pan out very well.
Also, pan out can mean “succeed”. For example, if being a musician doesn’t pan out, then I’ll become a businessman. Or, the new phone didn’t pan out – it failed to gain popularity.
Look into means to investigate or examine something. For example, I’m looking into buying a new car means I’m considering buying a new car and I’m investigating if it possible. Here are some more examples:
The police are looking into the matter – they are investigating it at the moment.
I’m looking into selling my computer. If I think I can sell it for more than £200, I’ll do it.
The company is looking into opening new branches in a number of different countries.
He’s looking into using the internet to do all his advertising.
Notice that looking into + noun and looking into + verb -ing are both possible structures. For example, in the following sentence a noun is used with looking into: The university is looking into the problem of the lack of computers in the library. But in this sentence, a verb is used in the continuous tense: The university is looking into enlarging the library. This is a very common phrasal verb.