–at least from my own experience in a cancer bio phd program…

• lots of public speaking. presentations galore! everything from lab meetings to class presentations to journal club to seminars to posters. it literally never ends. i think i give at least 1 presentation a week (and because of it my public speaking skills have improved immensely.)
  
• the amount of creativity involved. pure logic won’t get you anywhere; you need to be just as creative as artists, writers, composers, etc. those who never think outside of the box will never succeed in solving problems posed by science and health.
• failure is a normal occurrence. be it a 10-day experiment gone wrong, a rejected grant, stinging comments on your presentation skills–you really need unlimited patience, determination, and super thick skin to grow and thrive here. 
• gossip travels fast and everyone has a reputation. even before the end of my 1st year i already know whose research is going places, whose lab is running out of money; who the smart students are, who the lazy ones are; which professors and students are probably going to be kicked out in a few years, etc. science is a small world, and grad school is an even smaller one. 
• and speaking of reputations, being known as the “lazy student” is the absolute worst reputation to have. it’s okay if you’re not top of the class, or if your experiments aren’t working. but if you’re not a hard worker then wow… you might as well start looking for a new place to live.
• your undergraduate grades do matter. oops. but not in the way you think! they’re not a very important deciding factor for getting into grad school, but when you apply for fellowships and grants, the reviewers actually do look at your undergraduate grades to gauge your work ethics. I had a post-doc in a previous lab that got turned down for a grant because he had a C in Hebrew. Hebrew! it wasn’t even related to science! 
• it’s really difficult to find a thesis lab. this is probably the hardest part. you have to find a lab that’s doing research you enjoy, has people you get along with (especially the PI), and has money/room for you. it’s definitely tougher than it seems, and many labs these days just don’t have the funding necessary for grad students. many academic labs prefer to hire post-docs because they’re cheaper–and have more experience–than grad students. 
• you have to be a good writer. everything relies on your writing skills: your qualifying exams, your research articles, your grants, etc etc. your success as a scientist is practically 90% dependent on how well you can write scientifically. so, if you hate writing, maybe look for another field. 
• but! despite all this, grad schools have lots of resources to help you improve your skills, be in public speaking, or writing grants. it’s true that everyone here wants you to succeed and be the best scientist you can be. so there will always be classes and workshops that will help you read and analyze journal articles, or overcome your fear of public speaking, or become a better scientific writer. 
• not every PhD graduate needs to do a post-doc, and it depends on what field you want to pursue after graduate school. if you want to go into academics and become a professor or head your own research, then yes, a post-doc is practically required for you to build more skills and publish more papers. but if you want to work in industry, they accept masters and PhD’s right out of grad school. it really depends on you and your goals!
• you must publish papers. this is especially important for PhD’s (and is a requirement to graduate). published research articles are the only way anyone can know you were productive in lab. aim for at least 1 first-author paper. 
• it helps to be personable. you don’t need to be an extrovert, but you do need to be able to network, be a team player, and in general be able to interact with colleagues, professors, and other scientists. the most successful scientists are the ones who can form useful collaborations with others. no one can work well in a vacuum. 
• and lastly, it’s not that bad :) I know grad students have the stereotype of being frazzled and frustrated and way too busy all the time, but it’s the life we chose and honestly, we wouldn’t have it any other way. and i’m sure if you’re considering this path, it’s something you’ll be ok with too :)

(dear other colleagues–feel free to add more!)