Nice update! Sounds like things are progressing slowly but in a good way. The fact that you have had several brief conversations with her is encouraging. The next step would be engaging in longer conversations. I know you mentioned various obstacles: picking up your brother, next class starting, classmates in the way, and students physically in the way creating barriers.
If you can, see if you can meet her at the library or student center some time to study or prepare for a test or assignment in the class you have together. This will allow you more one on one time without as many distractions and ultimately more time. Keep the conversations at first to school and the task at hand. If she seems to be smiling or enjoying going over the study material possibly bring up something like... how did your weekend go? Did you catch the _____ game? Enjoying the fall weather, etc... she how she responds and what she might be interested in outside of your study matter for that class.
You can play it slow and continue with more study sessions or ultimately go for it and ask her out to do something that she may have hinted that she likes. Say watching a certain football team, going to certain style of concert, a certain type of food, or whatever it may be. Also don't rule out the idea of a group setting. Something where there would be a handful of people in a casual environment. Something like hey my buddy is having some friends over for the game. If you would like to come over and watch it with all of us you are more than welcome to. At this point if she says yes it will give you an excuse to ask her for her number and give her yours. You can tell her you will call or text her with the address and details etc...
I am not trying to be mean but if you really like her and don't move in a logical progression you will either be placed in the friend zone permanently or be too late to the party and someone else will have beat you to it. Most guys including myself have had this happen at one point or another.
Remember that should you happen to fail or find she isn't interested there are plenty more. Also it will help you as a learning tool for the next time you have to approach someone and try to progress things. I am rooting for you as is many others on here. Good luck with everything!