My car tire won't fill with air..?

You can replace the core your self not he entire stem A core tool will unscrew the old one out of the stem and replaced with a new one. The tire will go flat and need to be refilled after the new one is in.

I would think you could pick the valve core tool up and new core at a parts store. No, you do not need to take the wheel off

If you get the whole valve stem you will need to have that installed at a tire shop. You can try the stem/needle as you call it, but if someone has put SLIME in the tire before the whole valve stem may need to be replaced. Since you can't do that your self you had best off taking it to the tire shop and having them put in the new stem. Where I live the whole bill would be $9.00. If you buy like the stem needle they usually make you buy more than one at a parts store so the price would be about $5. There are a chain of tire shops on the west coast and they don't charge you for certain tire repairs. It is like a man's name.

The valve stem might be bad, but the bead might be broken too. Either way a tire shop is the best place to go. To replace the stem requires the bead to be broken. Years ago it was easier to get the bead to break and reset. But those times have changed, some times a device has to be put around the tire to help the bead to seat and getting them to break is now almost impossible too. One reason that they are hard to break is that there are sealants a lot of shops use to seal the area.

So take the tire off the car, support it some how (factory jacks just can't do the job) and run it down to a place that fixes tires.

It's a valve stem core and you can probably get one at WalMart. Like the other guy said put some water on the valve stem and look for bubbles to determine if it's leaking.

Shaking my head! Get a valve stem removal tool and a valve stem. Unscrew the valve stem using the tool and immediately screw in the new valve stem. You will need to put in air because you will lose some air when you unscrewed it. No need to take the tire off, no need to pay someone $9 to do it!

It is called a Schrader valve. It might be leaking, but they usually don't. Put some soapy water on it and see if it bubbles. If it does, you might only need the piece that goes inside, called the insert....but again I doubt it is the valve insert. Do the bubble thing.

to install a new stem you must unmount the tire from the wheel, just go to a shop and have them do it

Unless you have all the proper equipment at home (which I doubt you do), you'll have to take it into a shop and have it replaced.