First of all the main thing you need to do is BACK UP all your documents, pictures, etc....anything of value... on both computers. Just get an inexpensive external hard drive and copy to it.
That way if you do get some sort of malware you will be able to ditch those old computers and upgrade.
But yes, he is right, many viruses take advantage of outdated operating systems and their security flaws.
If you are happy dealing with Best Buy and want support from the Geek Squad, I suggest you go to a store at a non-busy time, and tell them what you need--a basic desktop tower. You should have several options under $500.
Schedule an appointment to have someone come and show you the ropes of Windows 10, and set it up the way you want it. It can be configured to LOOK a lot like your older systems but work MUCH better.