what is verbose mode

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That's a start mode that shows the log instead of the Apple icon

See Booting to and managing verbose mode in OS X | MacFixIt - CNET Reviews. Verbose mode simply displays what is going on during the startup process.

is there a way to actually have the steps that verbose mode takes you through to be saved in a file?

They are saved in a log file.

Open your Console Application (/Applications/Utilities/Console) and check the system.log and view boot sequence

Or, open Terminal and type:

sudo dmesg >> ~/Desktop/kernlog.txt

to save the contents of the boot up sequence from last boot until now to your desktop.

Thank you 😉