I don't have a PSP, but my Switch and DS are both connected online.
Switch shouldn't require anything fancy. WPA2 internet should work, or you can buy a LAN adapter and use ethernet. Most online games require a subscription.
DS is a bit more complicated. First of all, official servers are down, so you'll need to connect to wiimmfi's servers. Tutorial here: https://www.youtube.com/watch?v=Sh0AZR-tKwM&t=6s
The biggest hurdle is that the DS only supports either WEP or open internet, and it's unlikely that your current connection is either of those. You've got a few options:
1. Plug an older wireless router into your new router (this is what I did) and use the connection from that. Hold down the reset button on the old router to clear any previous settings/passwords, plug the port labeled "internet" on the old router onto any of the free ethernet ports on your new router, and you should be good to go.
2. Make a wireless hotspot with your phone and set the security settings to either open or WEP. This works on Android, but not Iphone since Iphone only lets you make a WPA2 hotspot.
3. Make a guest connection with no password/security. I don't know how to do this (I don't think my router even supports it) but I'm told it's an option
4. Buy a Nintendo Wifi USB adapter and plug it into a computer that is connected to the internet. Unfortunately, you'll need to be using windows xp for it
5. Disable security settings on your router. This is probably a bad idea.