Some terminology first.
- TUBE is the doughnut-shaped ring that holds the air (inner tube)
- TYRE/TIRE is the bit with the tread, and on your bike this holds the tube in place.
So you got several punctures in the tube caused by something sharp in the tire
You damaged the tire somehow and have a replacement. Now you've removed the sharp thing from the tire, and you want to know if the tube needs replacing ?
ANSWER Probably not. If the tube holds the air pressure its fine. If the tire has worn tread (ie to the point you're through the rubber and riding on the threads, or if it feels really thin to your fingers) or if the tire has a cut or significant hole, then replace it.
I'd recommend you throw a spare tube in your schoolbag, along with a mini-pump and tire levers if you need them, and a tool for your axle nuts if required. This will let you change tube on the side of the road, and patch it at home later.
Replacing the rear tyre should reduce the frequency of punctures too. The rear wheel suffers punctures about twice as fast as the front wheel.
Also, you might consider avoiding things that generate punctures too - which means not hopping kerbs/curbs, not bouncing through potholes, and going around patches of glass on the road/cycleway.