Whether your existing network is going to give you the performance you need / want is going to be highly dependent on the amount of data you are trying to send over it, so we can only give some general guidelines here.
Older network technology is pretty common in households who may not have upgraded their internet connection for some years.
Equally, it is quite likely that they started with a network that was standard technology at the time (say 100Mb ethernet and Wifi G or N) but have subsequently had an internet speed increase but (for example) still retained the existing wifi.
So, here are some simple guidelines to help you decide whether a network upgrade is needed and if so where and what…
I would expect that most people have a mix of requirements across their home:
A normal place for using your PC(s) - say your lounge or home office - and you might need to download large files from the internet or from your NAS.
One or more Smart TVs or streaming boxes in specific rooms, and you may have a 1080p or 4K or even 8K TV and you don't want the screen to freeze whilst it buffers or get pop-up messages (from e.g. Plex) saying that there isn't enough bandwidth to stream properly.
Internet access from their mobile phone or tablet wherever they are in the house. If you are watching video on a large-screen tablet wherever you happen to be, you may need significant bandwidth, but generally speaking your mobile / tablet bandwidth requirements are going to be lower.
You NAS may or may not need fast internet access, but if you are streaming media from it then it almost certainly will need fast LAN access.
Work out which areas of your home need fast access to the NAS or to the internet. For these areas (rooms) you will want (say) Gb ethernet / wifi AC, whereas (say) 100Mb ethernet / wifi N may be sufficient everywhere else.
If your media streaming is 1080p or higher, you should probably be thinking of Gb ethernet / wifi AC for those rooms needing this.
If your internet connection is (say) 60Mb or more, then you probably should be looking for Gb ethernet connection to the LAN from your router.
If you need Gb ethernet anywhere, then you will need a Gb Ethernet switch. Your router may have a Gb Ethernet switch inside if it has sufficient ports for your needs, otherwise you may need to buy one.
If you are having a Gb switch, then you almost certainly also want a Gb port on your NAS hardware.
These days, you should be able to see from hardware lights or from a web-GUI what speed each device on your network is achieving i.e
For wired connections, the connection between the switch and another device depends upon:
The capabilities of the equipment at either end. If both are Gb capable, then in theory the link is Gb capable.
The quality of the wiring. In theory you will need Cat-6 cabling for Gb ethernet, but in practice for the short distances you might have in your house, Cat-5 cables can still be fine, however in both cases the quality of how the RJ-45 plugs are fitted at either end (and the quality of the RJ-45 resistance to corrosion) can have a significant impact.
If your e.g. Wifi Access Points are PoE, then you will probably want a PoE switch which will be a little more expensive.