"Expecting ___" is used to talk about things that you think are going to happen in the future. Use "expecting ___" when there's something that you will be surprised about if it doesn't happen. For example:

Hey, where's Sammy? I was expecting him to be here.

You can talk about objects or events that you're expecting:

I'm expecting a package.

They're expecting rain tomorrow.

I'm expecting a phone call.

Or you can use the phrase "expecting ___ to ___" to talk about actions that you expect:

I'm expecting them to get back to me with an answer by the end of the week.