You could also use Bluetooth. I am sure the old E Machine doesn't have that capability but you can get a Bluetooth dongle (USB plug-in). Your new computer probably already has Bluetooth built in. Then transfer the files wirelessly.
big enough-- 32-64 GB. Use the windows backup to find all your important stuff, tell it to put it on the removable disk, then open the disk on your new computer and see what's there and copy it to where you want it to go.
We use the "Geek Squad" at Best Buy as there is more involved than simply moving files. They do our business desktops and laptops as well as personal computers & as I recall, they charge $90.00 at least that is the price here in this general area.
I wipe my computer whenever a major update of Ubuntu Mate comes out. I back my files regularly onto several thumb drives, you should be backing regularly too. Just back everything up on a usb drive and install on the new machine. Drives are dirt cheap, and will come in handy in the future unlike paying Geek Squad or paying for a duplicate Bluetooth.
Been almost a decade since I used Windows, but I'm sure you still (as in Linux) have to disable the USB port before pulling the usb drive out of the USB port. Should be a menu prompt. Or just turn computer off. If you want them out live you can ruin them.
I just did that. I have a Clickfree backup drive. I updated the backup on the donor. Plugged it into the recipient. Hit restore. In a few minutes (maybe 5 or 10) the transfer was complete. It even transferred my address lists, etc.
That would give you a good way to back up your computer as well. And that's a very good thing.
The only question concerns operating systems. That recent operation was from Windows 10 to Windows 10. It might be a little more complicated if you were going from say Windows 7 to Windows 10.
Instead of the USB stick, you could also use an external hard drive. Back up/copy your data to the hard drive, then install it on your new computer. 2-3 TB external drives are relatively cheap. You can use it later as a backup device to store your data in a separate place. I like external hard drives that don't have any proprietary encryption on them so you can more easily extract your data if the drive electronics crash on you. I can't remember the Western Digital model right now.
EDIT: LOL, I see I was beat to the punch. I question cloning the entire system if you are moving up to a much newer operating system, but my computer savvy is very low. I would just start from scratch and install new software if doing a big upgrade.
1) USB drives
2) take our your hard drive and plug into another computer
3) network them and turn on some sort of network drive sharing
4) download some sort of remote desktop software and transfer files over (i.e. tightvnc)
5) if it is a small file, just email
6) pay $20 to a teenager to do it for you.