I recommend you get any x86 with at least 2 gigabit ethernet ports and at least 2 cores preferrably 4. It makes sense to get something that has AES-NI instructions, like a Celeron 3xxx or 4xxx series, or an i3, or i5, if you can do it cheap enough. I particularly prefer Intel NICs. Most of these devices don't have built in switches they have individual NICs per port.
Next you'll want a smart switch, I'd recommend a Zyxel GS1900 series. The 24 port is about $100 on Amazon. Finally you'll want independent WiFi access points, the most common ones these days are either the TP-Link EAP series, or the Ubiquiti series, Ubiquiti requires you to drink more KoolAid I think (they have their own "controller" system that is absolutely required for proper function). The TP link also have a controller but are functional stand-alone as well.
If you're going to put up more than about 3 APs you might want to get yourself a Power over Ethernet switch, like an 8 port one from TP-Link or Zyxel to keep costs down.
Topology wise, I recommend you bond at least two NICs on the x86 and plug them to a LAG group on your switch, then set up two or three tagged VLANs, say 1 for LAN and 2 for WAN and maybe 3 for Guest if you want that, and maybe 4 for admin so you can do things like changing settings on the APs and lock out the LAN and Guest (and WAN!) and soforth from adjusting settings.
Plug the fiber into a port on the switch that's untagged for WAN VLAN and plug the bonded NICS into a LAG group that's tagged for all the VLANs and then tag the ports where the APs are plugged for LAN and GUEST (and whatever else, maybe ADMIN VLAN) and set up the APs to expect the tagged LAN and GUEST and ADMIN vlans... You'll have a great setup by the end of it.
Thank-you (and the others that replied) for this information. It is partially new to me and I will use it in a limited way in my small apartment setup.
However, what I am looking for is advice on what to ask the maker/supplier of the pc before I buy. For example, I have the (perhaps mistaken) impression that a switch chip is needed to support multiple VLANs on a NIC.
A NIC can support multiple vlans on it's own, and multiple NICs can be bridged together in software. Almost no x86 machines come with built in hardware switches, each port is connected to a NIC, so you should look for something with 2 or more Intel nics (realtek are ok but less good in some ways).
It is still useful to have a switch, so that's why I mentioned a zyxel or tp link web managed switch. You can use it to plug in access points, segment the network with vlans for hardware that doesn't handle vlans directly like printers or internet of things devices or even desktop computers, and do QoS prioritization on packets. You can also set up link aggregation which is a good way to use your multiple NICs on your x86 via bonding. Then you get both speed and redundancy advantages.
Which add a couple of extra network ports (and a console port) and work quite well in the role of router - they have some fairly noisy fans, but these can be swapped out, and the unit can be aircooled depending on application and retro fitting a heatsink.