You can do A LOT with an old pc.
Part one it still works
You want to upgrade your pc. I get that but don't toss your old one.
Install a lighter os
Linux is your friend if you have a 64 bit pc with at least 2gb of ram
What is Linux? Linux is a kernel - the part of the operating system that talks to the hardware - that is open source (anyone can modify and use it for free) and built on the UNIX's standards.
Because of its open source nature there are a TON of distributions (distro's) out there ranging from simple desktop use (Ubuntu, manjaro, elementry OS) to enterprise server use (Debian, RHEL, Cent os) to hobbyist use (Arch, Gentoo, stali)
So what distro?
If your computer is reasonably new (lets say a 1st gen i5 and 6gb of ram min.) then I would go ubuntu
If you have an older machine (or you just like how it looks) they go elementry os
If you have a REALLY old machine then go Tiny Core Linux
Desktop linux is very easy to install:
download the iso image and follow the instructions to burn it on to an USB/CD/Floppy
just adjust your boot device (check your computer's manual) to the USB/CD/Floppy with your distro's iso image
Wait for it to boot and follow the on screen setup
Once you get linux installed you should have a faster, lighter, and more flexible platform to work with. here are some things you could do:
File server you can use Linux to create a Network Attached Server (NAS)
Secondary or backup computer just log in and update every month or so and you have a secure, stable, and reliable secondary machine
Sell it, an old computer with linux installed can be sold for a good amount on ebay
Part 2 it's broken
Well, what's broken?
Computer turns on but no display
On a laptop the screen is most likely out. It can be replaced but it's expensive and on an older laptop not worth it. Plug in an external display and you can make a low profile desktop with a built in UPS (battery back up for desktops and servers) or set it up as a file server.
On a desktop your graphics card may be out this an easy and - if you don't about graphics quality - a cheap one. just by a new graphics card, open up the side panel, find your current card, unscrew it from the case CAREFULLY pull it out of it's slot, and CAREFULLY slot the new one in.
This is your BIOS (Basic Input Output System) telling you something is wrong. every system has different beep code look up yours and find the fix. It's most likely ram but it could be your cpu or a bad PCI card. all these parts can be easily replaces on 99% of desktops. on a laptop ram is really the only thing you can replace and even that is becoming less common
if you get nothing at all it could be a power supply your local geek hasa spare one ask him/her to test it in your machine.
Beyond that your machine is most likly beyond your ability to repair. reffer to parts