Welcome to the forum! Any Mercian that has served 30 years should be treasured, I think.
Much has changed during that time, but many of us here have updated bikes from the same vintage. So knowledge of how to get there is not an issue, what is perhaps more useful initially is to let you know how far one can go, and what might be involved.
You might want to read
this and
this to get some idea of what obstacles exist and what updating can involve.
It is most likely that the distance between the rear dropouts of your bike is 120mm, you have a freewheel block rather than a "cassette", and your wheels are 27" while road bikes today generally have 700c wheels.
Practically all current hubs are cassette based, for 8 speeds and above, and the road ones require 130mm rear dropout distance. If your Mercian is steel and not more exotic than Reynolds 531, then it can be cold set with care to accommodate such hubs if you wish. However if one is adopting a modern hub one might want modern 700c rim too (because 700c tyre/rim choice is vastly superior - 27" wheels were being phased out already in the late 80s). But since 700c wheels are a little smaller than 27" ones, new brake calipers with longer (+4mm) reach may be required.
There shouldn't be much difficulty in acquiring modern triple chainset, bottom bracket or front derailleur to suit. Similarly for rear derailleur or chain. What you might want to consider, however, is whether you wish to retain your current (friction downtube or bar-end?) shifters, or use modern brifters (integrated brake lever - gear shifters). If you want brifters, some consistency consideration in terms of number of speeds and make is necessary for problem-free indexing.
At the other extreme, if you are happy with just having gears lower than what you have at present, depending on the existing 5 speed block, you might be able to buy a new 5 speed freewheel with a substantially lower gear (i.e. a larger large cog to e.g. 32T or even 34T) without having to change hub/wheel/brake/frame spacing etc. In which case depending on the Suntour rear mech's wrap capacity, in addition to getting a longer chain you might have to replace it with one having a longer cage.
If you have further questions, please just ask.