Download MX from here (top link):
Download Links – MX Linux
Now, I'll try not to post a wall of text here, so let me give you a few links and a few terms or jargon to search:
As tkcartoonist recommends, Etcher, Universal USB Installer, Rufus.
Make Bootable USB from Linux | Category | Pen Drive Linux
Now if your old system is too outdated, you may have trouble with the above tutorials, so you may have to go the command line route:
dd command. (be ULTRA careful with your /dev/sdX letters and numbers to avoid mishaps!)
How to make a bootable USB from an ISO in Linux
How To Burn ISO Image To DVD And USB Using dd
or search "dd command write ISO USB"
Again, please be VERY careful with the dd command! Double and triple check the letters and numbers in /dev/sd[letter/number]. dd is nicknamed disk destroyer or data destroyer for a reason!
Next you want to either find out how to enter the BIOS on your new machine, or how to boot directly to external media from the boot screen, if possible.
Try keys Delete, F2 or F12 are the common ones. It may tell you briefly what key to press on the boot screen. You may get lucky and be able to press a key to boot directly from external media. If not, once in the BIOS, you are looking for the term "boot order" You want your USB drive to be at the top of the boot order list. Look for terms such as "USB drive" "external media" "external USB"
Search terms are "entering BIOS" and "change BIOS boot order" or "boot external USB" maybe with the manufacturer of your machine or any brand name displayed on the boot screen.
Once you have changed the boot order, look for terms such as "save and exit" "save and reboot" "save changes and reboot" either in one of the menus or maybe a keyboard key.
Once rebooted you will hopefully boot from your USB drive, ready to install.
Good luck!