It was 5:45 in the morning when I woke up, brushed my teeth, I took a shower, made a coffee and I went to the airport.
It may suggest that you have done all those things at once, or that you have done without any pause in-between. It´s my thinking here, though.
I would redo it making some slight changes:
It was 5:45 in the morning when I woke up, brushed my teeth then I took a shower. Next, I fixed my coffee and drank some of it and then I went to the airport.
Of course, there are other ways to skin a cat...but here goes my two cents´ worth on that one:
By lumping togheter the facts that -I woke, brushed my teeth and took a shower, I make it clear (or I suggest) that those are actions have happened in the same room or nearby, and that the actions happened not much as separate events. Denoting a sequence of sorts.
With the words next
I made a distinction from the previous sequence of events, to make it more "colourful" and informative I/you could also let the reader know "how the going to the airport" was. Say,...then I rushed to the airport (if I was in a hurry or late to go...).
It could also be a routine event: ...then, as I always do/then as I do everyday/then as I occasionally do, I went to the airport. Perhaps you went to the airport to pick up someone, or just decided to leisurely go there, or went in a given (or special day), etc...
These are just some comments on the fly, from a cursory reading, so other Forum participant might pick up where I left off, let´s wait for others to chime in.