What does 'Getting up comes easy to you' mean?

The full message is ‘Getting up comes easy to you, your muscles revitalized after your rest.’
Presumably, this means that I had a nice sleep, but I can’t think of anything different about tonight’s sleep compared to the times before. Is this due to something else in part? Does it really matter?

It’s just the vague ‘hidden health stat’ message you get when you wake up. If you have the Self Aware trait you can see your exact health stat, but without it you get these types of messages to give some indication of your immune system status.

How do you get a better or worse immune system?

The most common way is through your diet. Living off Pork Rinds and Atomic Energy Drinks is bad, Salads, Soups, and Nuts are good generally. Many foods are neutral. Radiation, infections, mutagens, and other effects can have pretty serious negative consequences also.

Okay, so basically just eat healthily rather than eating all that garbage around.
I typically don’t have my character eat much trashy food and drink for the most part since I assumed that that would probably be bad.

Pretty much. Junk food can be ok in moderation when you need raw calories. Your health stat always gravitates to the zero state. As long as your not getting 90% of your calories from junk food over a long period of time you won’t normally have any problems.