Wow, that's awesome! Mt. Rainier is a beautiful mountain! I live close enough to it too see its view (from the Canadian side of the border) and it is one of the more beautiful mountains in our area. You are so lucky!
You didn't say in your question whether or not your hike is a day hike or if you are scaling it all the way to the top (like the people going up Mt. Everest do). I found a website that will help you with what you must bring and what to wear, eat, have medically, etc. You are very tiny, so regardless of what you bring, MAKE SURE that your backpack has a waist-belt just below the frame (it's the one that ties around your middle at your hipbones) to help distribute the weight load evenly for you. Instead of your shoulders doing all the work, this belt will help your hips carry some as well - so less strain for you. If you are unsure about this type of backpack, ask the sales clerk at the gear shop exactly what a "hiking backpack" is. If they don't know, go to a shop that does. A "must" is proper foot gear. I've hiked several times in my life and the one thing I learned is that if your feet are uncomfortable, ALL of you will be too. It will make the trip miserable instead of fun. I recommend a good pair of hiking boots.
I hope your trip is enjoyable and the weather is beautiful. Oh, and remember to take a camera! You are going to see so much scenery and beautiful things to take pics of! Don't forget to take pics of your cousin and her boyfriend. Have them take pics of you too. REMEMBER TOO - if you make a campfire anywhere, make SURE it's out before moving on! I hope you have a blast!
(you are cool, girl - I LOVE Richie Sambora too... he'ssso sssexy)