What do I need to bring to basic training?

Your body and a positive attitude! The rest will happen.

Locks

I will tell you what I was told by my recruiter, who was a fairly good source of advice so i may have been lucky on that score.
Pack a small shaving kit, just the basics. A change of underwear and perhaps something to read in transit. Once you get there you will be too busy to worry about much of anything else. In processing is fairly intense and fast paced. (BTW if you do not have your SS# number memorized prior to in-processing you will have done so by the time it is over)
Like others have said once your first day is over everything you bring will be gathered and locked away until you leave. If you plan to write home a lot a good move is to buy some prepostaged envelopes and have your folks or a friend send them to you as soon as you have an address, by the time they get to you you will have your locker set up and have a place to store them.
Another thing to remember is that when you finish and they hand you back your belongings you will have pretty well packed your seabag/dufflebag and you will have to find room for those items. Things may have changed since my day but we went directly from boot to our next station so whatever we took to boot went with us to school and as such had to fit in our bag.
As a side note packing your bag is an art form and experience will teach you all the tricks to maximizing your storage potential but you will only have learned the basics of that art in boot.

You can bring what you want.but it will be taken if not needed while in basic training.kind of like going to jail,but you volunteered for the military.they will issue what you need. And I'm not saying the military is like jail,or am I lol. Just kidding. They will give you what you bring at the end of boot camp.

Shower shoes, towel, tooth brush & tooth paste would be #1. If you floss bring floss. Pack super light, your gonna get issued all the gear you need. Leave all your civilian possessions secured at home. If u bring phone it'll get taken and locked until your done. Don't bring anything you care about pretty much. Pictures etc leave it at home, you'll see it again.

The crazy thing to me with the US Military is they don’t give a dedicated list to new recruits. In the Australian Military you are given a list of required gear and paperwork to bring and told do not bring anything not on that list. Recruiting here seems very disorganized in comparison.

All you need is your paperwork and you. Everything you need they give you, so don’t bring more than a few bucks, no possessions at all, get a fresh haircut, clean shirt and jeans, that’s it. Nothing more required.

Literally only take the paperwork they gave you. You'll have to carry everything you bring plus everything you get issued and it's a pain in the ass. Save yourself the trouble

What kind of shoes should we wear to basic? Should we just wear our running shoes to basic to save room or can we just pack our running shoes? Why being running shoes if they make us get new running shoes when we get there?

When I went to bootcamp in '79 they had us put all our personal effects in a box at this warehouse where we would pick it back up after we graduate. I simply threw all my clothes in the trash.