Should I be concerned about the CPU usage (I am nervous because I only recently purchased the laptop)? I am nervous because I only recently purchased the laptop)
From a hardware (damage) perspective, your Mac and the CPU itself is designed to protect itself from overheating. It will kick in the fans, then throttle things back and finally, if it must, it will shut itself off. Your risk of doing damage to your machine is virtually nil.
From a software perspective, I wouldn't worry either. It sounds like your Mac is doing what it's supposed to do which is turn on the fans during high intensity CPU activity (compiling code) or watching video (GPU). That said, I would probably look at something else besides Chrome as it's a known resource hog and not very "kind" to memory or CPU utilization.
It's also important to remember that the 13" MacBook Pro doesn't have a discrete GPU so it's the CPU that's going to be doing all of the graphic rendering and it will use up to 1.5GB of RAM as video memory (Intel Integrated Graphics Chipset). So, if you're compiling code and watching HD or 4K videos simultaneously, you could be maxing out your hardware.
So, it is possible you undersized your Mac for your needs, so from a value perspective, I would defiantly look at that. If you're still within your return period, it's something you should make a priority to evaluate.