race against time = to hurry before a deadline; move quickly.
Racing against time, he got his coat and the newspaper and ran out to the train station to make the 10:15 train.
rain out = cancel because of rain or bad weather
Our canoe trip got rained out so we rescheduled it. (almost always used in the passive voice)
read (one’s) mind = to know what another person is thinking.
He knows her so well he can practically read her mind.
Can she read his mind?
right away = immediately; now
If he doesn’t get help right away, he’ll drown in the quicksand.
right on = an exclamation of happiness or approval; also means that something is correct or accurate.
"Oh yeah! Right on! I just found out that the stock investment I made yesterday doubled already!"
rule out = to say no; to determine that something isn’t true or factual.
Although he was jailed at first because of his suspicious activities, he was later ruled out as suspect in the crime.
The police ruled him out as a suspect.
run across = to find information; to meet someone or something suddenly.
Whenever my grandmother runs across an interesting passage in the Bible, she emails it to me.
run down = to review a list of things or read something quickly.
Let’s run down this list of candidates for the new sales position.
rundown = (adjective) a feeling of fatigue; tired.
Patricia has been feeling kind of rundown lately. It’s probably because she’s pregnant.
a. run into = meet or see someone after a long time away.
Fred ran into an old girlfriend at a birthday party last week. He hadn’t seen her in years.
b. run into = hit something, usually with a car.
When she ran into the other car, the windshield cracked in several areas.
This YouTube video provides more examples for using "run into."