I am graduating from undergrad this May and will be entering medical school this summer. Should I be doing anything to prepare?

First of all, Congratulations!!! Secondly, don't do anything! Enjoy yourself! If anything just prepare your living arrangements/financial details/etc. But don't study, enjoy your time! 

Is it okay if I don't apply for med school immediately after undergrad?

Plenty of people take time in between graduating from undergrad and applying to medical school to work, get clinical experience, travel, etc. It's not "negatively" looked upon when your application is considered, though it's a better idea to take that time to help your application, so you can keep your letter writers and experiences current for your application.

As for the break in school work and getting back into it upon starting med school, it can be a challenge for some, especially those who have come from jobs, but it can be done!

How important is shadowing a physician before I apply?

Shadowing is an extremely important portion of your application to medical school. It not only helps your application, but the experience will also help you writing your personal statement.

It's also a good idea to pick a physician and shadow them for a good amount of time, so that they can get to know you better and write a better letter of recommendation (if you ask them for one). Shadowing multiple people for short periods of time is probably not as ideal.

Do you have to have a science degree to get into med school?

Answer: Nope!

It's a good idea to take quite a few upper level courses in biology and chemistry, and do well in them. As long as you have the required prerequisites (which can be found on UNM SOM admissions page), you can major in any subject you'd like.

Just make sure to excel in the natural science courses that you do take.