What's the difference between first and at first?

First – At First

Use “first” to show the order of events. 
– First, I woke up. then, I made coffee. 
– First, plug in the computer. Then, turn it on. 
– Get to class first so you can get a good seat. 
– Print these documents first. After that, send them to the sales team. 

Use “at first” to show a change
– At first, I was really sleepy. But then I had some coffee and felt better.
– At first, it was really cloudy. Later in the day, it cleared up.
– I was really sad about my test grade at first but now I feel better.
– She had  a hard time in Spanish class at first but now she’s doing good.