Does it only wobble, shake and feel unsteady at speed? Or is it all the time and worse at 60+?
Does it go away over a certain speed too?
As a general rule, vibrations that only happen at speed are from un-balanced wheels and/or bad tracking. Vibrations that are there all the time tend to indicate worn suspension/steering bushes/balljoints, or warped brake discs.
Brake discs are an easy diagnosis. If it gets worse when you brake, its them.
Worn bushes etc can be spotted by jacking wheels up and giving them a good solid wobble. Any knocks or excess movement is a dead giveaway.
The most likely cause is balance/tracking though. With any old-ish car the tracking will probably have crept a bit over time so it cant hurt to have it done. Most tyre shops will do it for £20-£40 ish. Go for one that does laser tracking. If you ask really nicely they might throw in the wheel balancing for free too.
If not, its only a few quid per wheel, start with the fronts.
Before any of that though, check your tyre pressures