How do I increase the number of lines shown on the Ubuntu terminal window?

Click Edit→Profiles→Scrolling. Then increase the value under limit scrollback to: or untick it to allow scrolling through unlimited lines.

From comment by Kevin: Be careful with unchecking that option. If a program goes crazy and spews lots of data onto your terminal, you could have memory issues.

Open the profile settings for the terminal via

Edit > Profile Settings.

Or with a right click in the terminal:

Profile > Profile Settings

Change the value for Limit scrollback or remove the mark for unlimited scrolling.

Thinking about this in a different way, you could also redirect the output to a file and then use less +F to stream the file into less. Pressing ctrl + c will stop the stream, and typing shift + f will continue the stream.

This also gives you the added benefit of being able to highlight things you care about. For example, typing /my_search_term (forward-search) or ?my_search_term (backward-search) will also highlight my_search_term in the file. This highlighting will continue as new data streams in.

How do I output to a file?
There are a couple of options, but for completeness, this may be best:

start_rails_server &>> my_new_logfile

The &> will pipe both the stdout and stderr to the log file. (You can pick one or the other by using a number, 1 for stdout and 2 for stderr, though stdout is implicit so you could just do cmd > file and you would get stdout printed to your file.)

The angle bracket sends the output to my_new_logfile, and the fact that there are two means that it will append to the file if it exists, and create a new one if not. If you used a single angle bracket, then any existing file would be overwritten:

start_rails_server &> my_new_logfile