I found a photo of a Schwinn Trail Way hybrid on Amazon. From that photo, it looks like your bike has what is called a "threadless" stem. You won't be able to raise the bars straight up and down.

However(!), it also appears that your stem is adjustable, in that you can loosen a bolt and rotate the front part of the stem up or down. If you rotate the stem upwards, the bars will also get closer in, so two adjustments are really happening at the same time.

There should be a hex bolt on the side of the stem where it hinges. Loosen that. You may need to take it all the way out. I had an adjustable stem once that was designed such that it would not rotate unless I took the one hex bolt all the way, and then I had to pull off the side cap and slide the stem apart a bit on the other side. I'm not explaining that very well, but once you get the hinged part of the stem apart you'll probably see what I'm talking about. My stem was designed not to rotate unless the hinge portion was mostly taken apart.

Once you get the bar rotated upwards, you may find yourself wanting to readjust its angle. You can do that by loosening the bolts on the very front of your stem and rotating the bar to wherever it feels good. Tighten the bolts when you're done. Then you might want to adjust your brake and shift levers. To do that, loosen the bolts holding them onto the bar. Rotate them into a new position. Retighten the bolts (But not too tight, else you might break a plastic part. Just get them snug).

BTW, there are two bolts that attach your stem to the steerer tube. DO NOT loosen those bolts unless you are reasonably sure of what you are doing. There are one or two things that can go wrong when you loosen and retighten those bolts (as I've learned the hard way ).

If you truly want to raise the bars straight up without also bringing them in closer, then you need to either buy a new stem with a higher rise to it, or you can buy a new handlebar with a higher rise to it. I recently took the latter approach on one of my own bikes.

I don't know what to suggest about the rack that you are trying to mount.

I hope all this helps.